For the next four years, I will be attending the Colorado School of Mines, in Golden. Having grown up in the Bay Area, as well as having seen much of the Sierras, what is there to do (hiking, climbing, etc) within the vicinity of the campus or close enough for a college student without a car?

For technical climbing (sport + ice), look up clear creek canyon at mountainproject.com

As mconnell mentioned, check out mountainproject.com for climbing near Golden. Cycling on U.S. 6/Clear Creek Canyon is prohibited so you'll have to find someone with a car to get up that way; but there's lots of climbing at North Table Mtn accessible by bike.
Popular hiking and mtbing spots close to town (i.e. not requiring driving) include: White Ranch, Chimney Gulch, Apex, Matthew Winters, Dakota/Dinosaur Ridge. Note that Jeffco Open Space has recently placed restrictions on the direction that mtbers can use the trails at Apex to minimize conflicts b/w different users. Check out their website for details.

On the west side there are many hiking trails, some with some nice elevation gain. You’ll be able to walk to a few. Make friends with someone who has a car and your options will go up a lot. Denver Metro is not a great place to get around without a car. We have a local transit system but our population density (lack of) doesn’t support frequent runs everywhere.

Trails Illustrated – Boulder Golden - Map 100 gives a good overview of hiking trails in that area.

There is a lot to do in Golden, however going to school of mines you will have time for absolutely none of it Check out the micro-brews in the area, lots of good beer. Ill second table mountain, you should be able to find some good rock within a bike ride. Good luck with either engineering or geology (Im geology myself at CSU).

I graduated from CSM in 1980, so any info I have would likely be outdated. But when I lived in the basement at 1717 East St, I could go out the back door and hike straight up to the top of South Table Mtn. Or when feeling less ambitious, go a couple blocks down the hill and take the "short tour" at Coors. It was also possible to get up on top of South Table Mtn with my jeep, and blast around the old dirt roads up there until we ran out of either gas or beer.

I'm from the Bay Area and I'm attending CU Boulder, gonna be a junior next year, so I was in the same boat as you. You'll figure out stuff to do pretty quickly, but my advice to you is to just get a car. I didn't have a car my freshman year, and even though I was able to find people to get to the mountains with, it was still a pain in the butt. And IMO, public transit is not that great around (maybe living in Golden makes this different that living in Boulder). If you have your own car you have the freedom to go wherever, whenever, though paying for gas is a bitch, it gets you to the mountains. Find friends to split gas with, and it's really not bad at all.