General suggestions: Avoid anime that's longer than 20-50 episodes or so. Avoid anime that is still on-going (IE: Mega series like One-Piece and Bleach that are reaching their 250'th episode and then some tend to grind on the nerves.) Oh, and while there's been several spin-offs of digimon since the original came out really they never managed to re-capture what made the original series so good. Generally when they take the same setting, and simply insert new characters its because the old one was at a dead end.
Anime anyone can get into
Cowboy Beebop: It's not set in the typical anime drudge that saturates the market. Characters are believable, the plot line is believable, the style of animation sits somewhere in between western and eastern, and it's just generally all around "good." It's probably a classic too, if anyone hasn't already pinned the name to it. Watch it, you won't be disappointed. It's also notable because it's one of the few animes that actually has an English dub that can be described as something other than "bad" or "tolerable."
Foolie Coolie (FLCL): This is the show I show to anyone who says they want to try some more comedic anime. It's a good primer on the "WTF!?" factor that's in a lot of it, and at only 6 episodes if you really dislike it that much it's over before you know it. Probably doesn't hurt that below the frivolous balls-to-the-walls humor there's actually a pretty well played out theme.
Ghost in the Shell: It's... well it's generally good. There's some hit-or-miss factoring in it but over all the good parts more than make up for it. Don't watch the second movie unless you're really die-hard about it. Actually, a good place to start would be the first Stand Alone Complex (GitS: SAC) of the two animated series. The plot has no bearing on the movies (and visa versa) and generally it minimizes the drawn out moral dilemmas that they endlessly discuss in the films to the point that I literally fell asleep watching the second movie my first time through.
Trigun: Cut from the same basic stylistic preferences that make Cowboy Beebop stand apart from your typical fair, Trigun is another series that can just, in a word, be called "good."
Xam'd: The Lost Memories and / or Eureka 7: I like Xam'd more just because the protagonist is more likable and the series avoids the mood swings rather well, but the two are incredibly similar at a basic level. Watch one, watch both. I don't really care, they're both great. Regardless, they're probably the best of the lot for anime that mixes robots, action, and romance.
Tenngenn Toppa Gurren Lagan: Yeah. I probably spelled some part of that wrong. Google Gurren Lagan if you really have to know. It's credited as the series that pulled the anime industry out of the depressive slump and the "oh god, we don't want to touch that with a 5 foot pole" effect that Neon Genesis Evangellion started. It's kind of silly, but it tells you pretty forthright that they're not trying to be super serial.
Full Metal Alchemist: Its good. End of story. Watch Brotherhood rather than the original. It sticks with the manga (Since the original anime finished before the manga did!) and it actually explains a lot of would-be plot holes.
For more advanced viewers...consider yourself warned!
Neon Genesis Evangellion: Its the primer for the genre of anime that inevitably analyzes the psych of the characters. First, no, there's no deeper meaning. The Christian / Kabalistic symbolism was literally only thrown in to look deep. Second, this is not something to watch quickly. I was skull fucked for a week or two after trying to do it, and other people who tried the same agree- do not do it. Oh, and be prepared to do a lot of extra-ciricular viewing. The original series got thrown into a proverbial shit bin at the end as the studio drew flak for the nudity content in it (this was before Japan had more segregated viewing on the air) and as the budget ran out, and the director literally had to go be put on Prozac they went for a head banger ending that....well. Makes you want to bang your head. Watch the movies (I'm probably mixing it up, but Death is a colossal waste of time as it is literally the series compressed into a single movie- watch it if you need the re-cap, or if you're really that hardcore. The ending credits are some of the best in anime, while Rebirth is the actual plot advancer.) From there there's all kinds of fan fics, and series reboots you can watch. Wikipedia it if you're that curious. Personally I highly recommend the Re-take manga. Just try to find the all-ages edition.
Elfin Leid: This is actually something you probably shouldn't watch unless you want a primer on the excessive gore genre. It's not actually a good anime, and even the guy who came up with it hated where the anime went (it was a different director, mind you) which should be the first hint that it's a train wreck waiting to happen. What you get is a anime that's mentally all over the place. It goes from women literally getting diced up, in all the gory details to frivilous fan service and a character with two personalities, one being basically a homicidal maniac, and the other being.... basically a cat. I'd tell you to just read the manga if you insisted on this one.
Ergo Proxy: Its actually almost on the first list, but it's still heavily mired in the psychological stuff that turns off a lot of people, and even after watching it twice a lot of it escapes me. While the series is good, it's the kind that after each episode you have a paragraph of external reading to do so that you pick up what you missed.
Fate / Stay Night: It's not considered the best visual novel ever for no reason at all- it's actually pretty damn good. The only thing that may draw people away from it is the sheer number of scenarios and differing plot lines.
xxxHOLIC: erm.... where do I start with this one? Don't get me wrong, its a good one, for sure, but it's odd really. Its an experience really, and it's a fun modern twist on the analysis of human vices versus virtues.
Honestly, I could go on, but this is a pretty basic list for what you could get started with. If you really want a complete one go to 4chan, take everything you read with a grain of salt, and wait on the anime general board for a recommendation thread to roll around. Usually they'll surface if you don't mind sifting through the usual shit that surfaces on 4chan.
Anime anyone can get into
Cowboy Beebop: It's not set in the typical anime drudge that saturates the market. Characters are believable, the plot line is believable, the style of animation sits somewhere in between western and eastern, and it's just generally all around "good." It's probably a classic too, if anyone hasn't already pinned the name to it. Watch it, you won't be disappointed. It's also notable because it's one of the few animes that actually has an English dub that can be described as something other than "bad" or "tolerable."
Foolie Coolie (FLCL): This is the show I show to anyone who says they want to try some more comedic anime. It's a good primer on the "WTF!?" factor that's in a lot of it, and at only 6 episodes if you really dislike it that much it's over before you know it. Probably doesn't hurt that below the frivolous balls-to-the-walls humor there's actually a pretty well played out theme.
Ghost in the Shell: It's... well it's generally good. There's some hit-or-miss factoring in it but over all the good parts more than make up for it. Don't watch the second movie unless you're really die-hard about it. Actually, a good place to start would be the first Stand Alone Complex (GitS: SAC) of the two animated series. The plot has no bearing on the movies (and visa versa) and generally it minimizes the drawn out moral dilemmas that they endlessly discuss in the films to the point that I literally fell asleep watching the second movie my first time through.
Trigun: Cut from the same basic stylistic preferences that make Cowboy Beebop stand apart from your typical fair, Trigun is another series that can just, in a word, be called "good."
Xam'd: The Lost Memories and / or Eureka 7: I like Xam'd more just because the protagonist is more likable and the series avoids the mood swings rather well, but the two are incredibly similar at a basic level. Watch one, watch both. I don't really care, they're both great. Regardless, they're probably the best of the lot for anime that mixes robots, action, and romance.
Tenngenn Toppa Gurren Lagan: Yeah. I probably spelled some part of that wrong. Google Gurren Lagan if you really have to know. It's credited as the series that pulled the anime industry out of the depressive slump and the "oh god, we don't want to touch that with a 5 foot pole" effect that Neon Genesis Evangellion started. It's kind of silly, but it tells you pretty forthright that they're not trying to be super serial.
Full Metal Alchemist: Its good. End of story. Watch Brotherhood rather than the original. It sticks with the manga (Since the original anime finished before the manga did!) and it actually explains a lot of would-be plot holes.
For more advanced viewers...consider yourself warned!
Neon Genesis Evangellion: Its the primer for the genre of anime that inevitably analyzes the psych of the characters. First, no, there's no deeper meaning. The Christian / Kabalistic symbolism was literally only thrown in to look deep. Second, this is not something to watch quickly. I was skull fucked for a week or two after trying to do it, and other people who tried the same agree- do not do it. Oh, and be prepared to do a lot of extra-ciricular viewing. The original series got thrown into a proverbial shit bin at the end as the studio drew flak for the nudity content in it (this was before Japan had more segregated viewing on the air) and as the budget ran out, and the director literally had to go be put on Prozac they went for a head banger ending that....well. Makes you want to bang your head. Watch the movies (I'm probably mixing it up, but Death is a colossal waste of time as it is literally the series compressed into a single movie- watch it if you need the re-cap, or if you're really that hardcore. The ending credits are some of the best in anime, while Rebirth is the actual plot advancer.) From there there's all kinds of fan fics, and series reboots you can watch. Wikipedia it if you're that curious. Personally I highly recommend the Re-take manga. Just try to find the all-ages edition.
Elfin Leid: This is actually something you probably shouldn't watch unless you want a primer on the excessive gore genre. It's not actually a good anime, and even the guy who came up with it hated where the anime went (it was a different director, mind you) which should be the first hint that it's a train wreck waiting to happen. What you get is a anime that's mentally all over the place. It goes from women literally getting diced up, in all the gory details to frivilous fan service and a character with two personalities, one being basically a homicidal maniac, and the other being.... basically a cat. I'd tell you to just read the manga if you insisted on this one.
Ergo Proxy: Its actually almost on the first list, but it's still heavily mired in the psychological stuff that turns off a lot of people, and even after watching it twice a lot of it escapes me. While the series is good, it's the kind that after each episode you have a paragraph of external reading to do so that you pick up what you missed.
Fate / Stay Night: It's not considered the best visual novel ever for no reason at all- it's actually pretty damn good. The only thing that may draw people away from it is the sheer number of scenarios and differing plot lines.
xxxHOLIC: erm.... where do I start with this one? Don't get me wrong, its a good one, for sure, but it's odd really. Its an experience really, and it's a fun modern twist on the analysis of human vices versus virtues.
Honestly, I could go on, but this is a pretty basic list for what you could get started with. If you really want a complete one go to 4chan, take everything you read with a grain of salt, and wait on the anime general board for a recommendation thread to roll around. Usually they'll surface if you don't mind sifting through the usual shit that surfaces on 4chan.