The choice to go with a heart monitor is sort of a separate issue. Without a training plan you may or may not find it useful.

As for wired vs. wireless, wired is a bit more of a pain to set up and some consider the wires unsightly, but they ARE more reliable.

Cateye and Sigma make some very good, reasonably priced computers. At the entry level you need speed, distance, cadence, and average speed. As you gain experience you'll sort of figure out on your own what other features you want.

Congrats on the new bike, and welcome!