How can I improve a Zombie deck to work faster?
You're a newbie that wants to improve on a deck so I won't go into deck building theory or anything like that, I'll just try to be useful and point out a possible approach (which is not the only one, by all means).

You have a nice zombie theme going on. I think you can go for a nice aggro deck with cards from the last couple expansions (Innistrad and Dark Ascension).

First of all, the focus. How do you kill? I'd say zombies are good at swarming. Let's go for overwhelming the opponent with a lot of creatures: Diregraf Ghoul, Gravecrawler are your all stars, so ideally you'd play 4 of each! Highborn Ghoul can be interesting for its evasion, Black Cat is also cheap and provides value when it dies and Geralf's Messenger is nuts although it costs too much black mana (so you just want 2 or 3).

Let's not forget some blue zombies as well! Diregraf Captain is pretty neat because it pumps all the other guys (4 of him as well), Stitched Drake is really powerful and cheap but once again, costs two blue (maybe play only 3)!

About your spells, I think they're all over the place! I'd just focus on getting rid of opposing creatures so that your zombies can hit.

Undying Evil is a neat trick to save one of your guys (plus it plays really well with Black Cat or the Captain). Other than that, you want Tragic Slip, Dead Weight or Victim of Night. If you can get your hands on other expansions I'd say Go For The Throat, Doom Blade and Dismember are all very powerful standard legal removal spells!

Finally, since you're playing blue, maybe some Ponder and Forbidden Alchemy would be good to trim your deck and get the good cards out!