Congratulations! (A Trek 5000 was my first 'adult' bike too.)

I agree with the other comments- the following are pretty important:

-Bike shorts (make sure they fit well or they may rub) - initially 1 or 2 pairs is enough (you can wear them under tights if you ride in colder weather).

-Gloves

-Some way of inflating a tire. I carry a small pump (it attaches to the water bottle cage). You'll want something that's always on the bike (otherwise the day you forget it will be the day you have a flat).

-Spare tube(s), patch kit (either 2 tubes or 1 tube and a patch kit will get you through most situations). Also tire irons.

-Hex keys (AKA allen wrenches)- these are needed for adjusting seat height, etc. (even if you have the seat adjusted properly, I've had very minor accidents (eg I fell off the bike) where the seat gets knocked out of alignment and I need to readjust it. There are some inexpensive tools (Park tools etc- they look like swiss army knives) that combine several hex keys of different sizes with screwdrivers (the screwdriver isn't absolutely essential, but is the sort of thing that can come in handy).

-Small saddlebag- while you can argue that this isn't essential, I think it's best to have all the aforementioned items always on the bike (otherwise you'll have forgotten them when you most need them). I also like to keep a $20 in the saddlebag so I'll always have cash handy.

A computer is a great thing, but not essential. If you're potentially riding after dark, at least get a flasher to hang on the saddlebag (this should cost less than $10).

Have fun!