Good-quality chain tool (I like the small Park Tool model, but make sure to tighten the pin with pliers and/or Loctite it in place)
Folding hex-key set such as Park Tool AWS-10
Perhaps some pliers
Spoke wrench that has the proper size for your spoke nipples
More deluxe stuff:
Pedal wrench is often nice to have
A chain scrubber would make sense for a 'cross bike especially
I like the Cable Luber for flushing dirt and water out of your shift/brake housings
Crank extractor (there are different models depending on your bottom-bracket spindle type)
Long 8mm hex key for reinstalling cranks
Chain whip and cassette-lockring tool so you can remove your cassette from the rear wheel
Repair stand, a big help for repairs or just cleaning
I would collect the tools as you need them. If you are new to cycling, you really wont need a crank-puller for a long while.
Avoid cheap bad tools such as pressed steel cone wrenches. Buy good quality forged ones. I get a lot of tools at junk shops and garage sales. Good tools stay good for years.
To start with, you need tools to fix a puncture, and a set of allen keys to ensure bolts are tight.
I woke up this morning and realized that a 'cross bike might mean a cyclocross bike (with drop handlebars like a road-racing bike), or it might mean a hybrid (mountain-bike-style handlebars). Some of my suggestions were centered on the cyclocross assumption.