Let me start by saying it has been a LOOOONNNGGG time since I've ridden a bike, but I'm trying to get back into it to hopefully shed a few pounds and spend time with friends while doing it! And this site is AWESOME!
I just acquired a 2010 Trek Valencia (FOR FREE!!). It is still rolling on the stock wheels and tires, which are Bontrager Race Lite Hard Case tires on Bontrager Satellite Nebula rims. The tires are in descent shape, but I would like to get something with a little more tread. I was also wondering if I could go any wider than 32mm with the current rims? Could I get away with 35mm or even 40mm tires? As I said before, I'm a bigger guy so more rubber touching the ground would make me feel better and I'm not overly concerned with super low rolling resistance as I don't plan on doing a long ride for a while.
Your rims should be able to accommodate wider width with no problem, but it's gonna be the frame/fork clearances that will dictate how wide you can
I would take it to your LBS and ask them. I'm a big guy also and I really love Schwalbe Big Apple tires. I have them on five bikes and on a tandem as well. 2" and wider are really nice if your bike will take it.
Welcome back to biking! You got a nice bike at an even nicer price. While bigger tires can certainly be more comfortable in some conditions they may not be needed. Where do you think most of your riding will be. Paved path? Proper inflation front and rear is very important to comfort, maintaining control and traction. The pdf below gives the correct pressure front and rear for a variety of tire sizes and rider/bike weights. A more aggressively treaded tire could actually be harder pedaling while not improving traction.
Good advice here: go to LBS/get suggestions.
One thing (referred to above): unless you've some intention of riding on unpaved/mixed surfaces, do not purchase tires with "tread". You are actually not only more efficient without on paved surfaces, but safer (more grip), especially when cornering and braking. That goes for wet conditions as well as dry.
Stay with 'slicks' unless you contemplate significant time on grass, dirt, gravel etc.