Get one of the amp/effect/speaker simulator boxes. They have headphone outputs as well as line outs you can send to a power amp, a stereo or your computer.
I bought the Johnson J-Station for $150.00 bucks. Its totally amazing. You will be inspired by the sound! Its like having thousands of dollars of amps ,speakers and effects rolled into one.
Another really cool device is the Korg Pandora. A friend of mine brought two of them over the other day. It has built in drum and bass grooves you can jam with. We plugged two guitars thru them and into a baby grand digital piano for amplification. Awesome! Theres a bunch of other brands, POD and some others.
I've been playing for 30 years, these little boxes come close to the real thing. Have fun, the bass guitar is great.(I always wish I'd played it as my main instrument instead of guitar).
Tommy P.