This is my backlog.
What should I not waste my time with?
It's hard to say, since we don't really know what kind of shows you would like, to be honest. For example, some people will say "don't watch that it's shit" -- but it's only shit from their perspective because they think that all moe is shit, when in actual fact if you like moe then you should probably watch it. Personally, I think that you should at least try to watch all of the shows on your list. Experience is always good. It'll help you find and develop your own taste in shows, and, with enough experience, you'll be able to determine how a show will turn out after just watching a few minutes of the first episode, or reading a synopsis of the plot. If you don't want to waste time downloading all of those shows, you can always just stream the first episode or two (from kissanime.to or twist.moe), or just torrent an individual episode, which wouldn't take very long with a decent internet connection.
Help me out.
And can anyone tell me if it's a good idea or not to watch Urusei Yatsura 2: Beautiful Dreamer having seen nothing of UY before?
I didn't watch all of them, but most of them are okay or even good.
Angel Beats, Fate UBW, Katanagatari are good, among others.
Watch at least a few episodes of Strike Witches (witches that use magic and normal weapons and mechanical parts to fight against aliens) and The idolm@ster (idol show) to know what they are about.
You need to be Japanese to understand Joshiraku.
Gate (japanese army in fantasy land) was pretty meh, but it's airing this season, so you may want to watch it to be able to follow it now.
I didn't watch many of these, sorry.
/a/ liked Gankutsuou and Hyouge Mono is even in one of the anime of the year chart (but I'm not sure if it's trolling or not).
Memories is a short collection and you should be able to watch it when you have time.
Natsu no Arashi is about a summer story with ghosts and time jumps. It has Chiaki Omigawa and Ai Nonaka in the cast, they need more roles, dammit.
Real Drive is about plumpy girls helping an old man solve problems in the virtual reality.
In order of preference:
Shimoneta (Funny as fuck, and *almost* has a message)
Melancholy of Haruhi (Classic)
Meme la meme (Meh, animation's fun)
JoJo: Battle Tendency (not muh thing, but if you like ultra-violence)
Chaika (This just sucks)
From most enjoyed to least enjoyed.
Anything below is not worth your time
Bokura wa Minna Kawaisou
Beyond the Boundary
The only thing worth your time is Haruhi.
Watch School Rumble and read the mango, they're both great and really funny.
You might also look into Chaika, Ergo Proxy, Food Wars, Umaru, Jormungand, and Idolm@ster.
Read Robotics;Notes and play NepNep.
My goodness. Well I'll pick a few I like.
RiN - I really enjoy this one. If you've read Beck it's basically more of the same except with a supernatural aspect that occasionally gives it a nice creepy atmosphere. Long chapters so the monthly update is always satisfying when you're caught up.
Getenrou - Starts off with low-stakes episodic mysteries with a comedic tone before weaving them all together to tell a more dramatic story. If nothing else it's unique.
Oofuri - The baseball scenes are very realistic with surprising tactical depth and the character interactions are always rewarding during the slice of life stuff. Consider watching the anime and picking up where it leaves off because it's a very faithful adaptation and the manga's art is only decent.