Check out Caroline’s. Also if you are planning on the comedy cellar try and go to a late show on a Tuesday or wed. That is when you’re more likely to see the best talent and even surprise celebrity guests. Very cool place.

I'm planning a NYC trip in 2 weeks. The Stand is also a good club. Not from NY, so take my words as you will. As for pizza, I wanna try a slice from Vinnie's Pizzeria in Brooklyn. They've been on Man vs Food and they have a pizza box pizza that I want to try.

I’ve only been to one comedy club in NYC, it was called Gotham and I had a really good time there! You should totally check it out.  

As for pizza, I am not the expert but a client suggested Di Fara to me a while back and I am planning to go tomorrow or Friday while I am visiting!

Downtown Greenwich village you can do both! (my hood!)
Go to McDougal Street between West 4th St and Bleecker, this is where the Comedy Cellar is, Standup NYC  is down the block and there is one other I cant remember. Then around the corner across Sixth Ave on Carmine is Joe's Pizza for the best slice in NYC!
Enjoy.

Seconding Priya on Gotham -- it's a good club, and big comedians sometimes drop in there unannounced.  Then again, they do that at the other places too, and probably more often at the Comedy Cellar.  You can't go wrong at either place for traditional standup comedy, and there's good pizza nearby, too:  John's of Bleecker Street is just a couple of blocks away from the Cellar, and if you want to be really lazy or are a comedy nerd, you can go to Ben's Pizzeria, which is just up the block from the Cellar and is the place where Louis CK wolfs down a slice in the opening credits of "Louie."  (I know we can't mention him anymore... sorry.)  Grimaldi's is a good pizza place near Gotham.

If you like alt-comedy or whatever you're supposed to call it, Creek and Cave in Long Island City is pretty cool, as is the Bell House in Brooklyn, which you could go to on the way to or from Di Fara's if you decide to go there for pizza.  Have fun!

Joe's Pizza by NYU is great.  John's of Time Square is also great.  Angelos Coal oven is my favorite.