I've been here 12 years, moved around about a half dozen times and never lived in the same place twice... well I guess that's technically wrong as I lived in "Denver" twice. Still, Denver Metro is huge and stretches for about 60 miles. It's one continuous town from about 136th to Lincoln which if that were a numbered street would probably be in the -120th range. So 'Denver' stretches for a ways.
But I'll help you out. Mid to late 20's. I assume no job as if you did have one you'd want to live close to that. $800 per month is a little low for Denver. You may be in a sh|+-hole with only $800 per month. The cost of living is higher out here (but most jobs pay more so it all cancel's out).
So I'd be looking in downtown, Capitol Hill (close to downtown), Washington Park (Wash Park for short {may be a little out of your price range}). D.U. area (Denver University {probably more in your price range}). And maybe the Highlands. Some links to the areas are below. Scroll around a bit and you'll see that all these places are relatively close to one another. Downtown is the furthest North. Highlands is slightly south and to the West. 6th & Pennsylvania is back to the east and slightly South. Wash Park is a little bit further South. D.U. is just south of Wash Park.
All those areas are somewhat close to downtown and the light-rail stations. There will be plenty to do, plenty of people your age, plenty of bars and restaurants, pot-shops and the likes. You can probably move into some apartments around 6th & Pennsylvania for about $800 per month for a 2 bedroom.
Stay away from the other surrounding suburbs as they are probably not what you are looking for (even though some may be a little cheaper): Aurora, Arvada, Broomfield, Boulder, Superior, Louisville, Lakewood, Highlands Ranch, Thornton, Northglenn and many others that I can't think of at the moment.
Yahoo Answers does not like the length of my sources so I'm just going to paste them here:
6th & Pennsylvania:
http://maps.google.com/maps?f=q&source=s_q&hl=en&geocode=&q=Denver&sll=40.23841,-76.136176&sspn=0.011269,0.070467&g=6th+pennsylvania+Denver&ie=UTF8&hq=&hnear=Denver,+Colorado&ll=39.725746,-104.981165&spn=0.000573,0.001101&t=h&z=20
Wash Park:
http://maps.google.com/maps?f=q&source=s_q&hl=en&geocode=&q=Denver&sll=40.23841,-76.136176&sspn=0.011269,0.070467&g=6th+pennsylvania+Denver&ie=UTF8&hq=&hnear=Denver,+Colorado&ll=39.700583,-104.968228&spn=0.018358,0.035233&t=h&z=15
D.U.:
http://maps.google.com/maps?f=q&source=s_q&hl=en&geocode=&q=Denver&sll=40.23841,-76.136176&sspn=0.011269,0.070467&g=6th+pennsylvania+Denver&ie=UTF8&hq=&hnear=Denver,+Colorado&ll=39.677797,-104.959989&spn=0.018364,0.035233&t=h&z=15
Downtown:
http://maps.google.com/maps?f=q&source=s_q&hl=en&geocode=&q=Denver&sll=40.23841,-76.136176&sspn=0.011269,0.070467&g=6th+pennsylvania+Denver&ie=UTF8&hq=&hnear=Denver,+Colorado&ll=39.747528,-104.995297&spn=0.004586,0.008808&t=h&z=17
Highlands:
http://maps.google.com/maps?f=q&source=s_q&hl=en&geocode=&q=Denver&sll=40.23841,-76.136176&sspn=0.011269,0.070467&g=6th+pennsylvania+Denver&ie=UTF8&hq=&hnear=Denver,+Colorado&ll=39.746975,-105.021422&spn=0.009173,0.017617&t=h&z=16