Depending on how many days you have to ride I would suggest these trails, starting east of Chattanooga and going west, with Raccoon being the closet to downtown Chattanooga and being the "must ride" trail of the bunch.
Day 1
Tanasi Trails - 75 miles miles of downtown Chattanooga - 32 miles of trail - camping:
http://www.sorbachattanooga.org/trails/trail_guide.php?state=TN&trail=2
Day 2
Chilhowee Mountain Trails - 70 miles east of downtown Chattanooga - 25 miles of trail - camping:
http://www.sorbachattanooga.org/trails/trail_guide.php?state=TN&trail=8
Day 3
Enterprise South Nature Park - east side of Chattanooga - 11 miles of trail:
http://www.hamiltontn.gov/esnp/
Day 4
Raccoon Mountain - west side of Chattanooga - 20 miles of trail - camping near by:
http://www.sorbachattanooga.org/trails/trail_guide.php?state=TN&trail=120
Day 5
As already mentioned, Monte Sano State Park & Land Trust - 80 miles southwest of Chattanooga - 50 (?) miles of trails - camping:
http://www.sorbachattanooga.org/trails/trail_guide.php?state=AL&trail=28
As far as places to drink.... Big River is not as great as it used to be. They changed their recipes when the merged with Rock Bottom and again when they merged with Gordon Brisch. While everyone will have their own opinion, I believe their product has suffered greatly. Drink more than two beers and you will likely wake the next day with a headache. Sad as this used to be my favorite place to drink and I live within walking distance.
For great beers I would suggest The Terminal Brewhouse -
http://terminalbrewhouse.com/ . They are located next to the Historic Chattanooga ChooChoo, only about 15 blocks from Big River. They have a great selection of beers, usually 6 different brews on tap at any given time. Their standouts are their Old Baldy (Bitter - seasonal so it may not be on tap during your visit) and their Magnum PA which is simply one of the best brews in the IPA category. They also have very notable Oatmeal Stout. The drawbacks are this place can be very crowded around dinner time, limited parking in their parking lot requiring you to park on the street and walk a bit, and the food is simply average. Side note --> their beers run on the strong side so be careful on how many you consume if you plan to drive.
O'Heiney's -
http://www.facebook.com/pages/OHeineys/189445954413498 - this is a local watering hole which recently opened. They serve the same beers you can find in most bars, the exception is they usually have the Nashville brewed Yazhoo Hop Project on tap. As far as I know they are the only pub in Chattanooga that is currently carrying the Hop Project. That aside, this place is owned by a mountain biker, most of the people behind the bar ride bikes, and many of the customers ride. This place is located round back of Joe Friday's Alaska Coffee house, you have to walk down the alley between the Joe Friday's and the law office next door to get to the entrance. There is nothing too special. It is just a fun place to hang out.
Moccasin Bend Brewing Company -
http://www.bendbrewingbeer.com/index.htm - is a nanobrewery. That means they produce very small batches compared to most microbreweries. This place looks kind of scary when you walk in, the bar area leaves something to be desired, but the brews run from amazing to out right strange. This is the type of place you go for the product, not so much for the setting or the crowd. They usually have at least one brew on tap made from something you would even consider making a brew from, such as their sweet potato brew or juniper brew. They are like a bunch of amazing home brewers that refuse to grow up. They usually have 10 to 12 different brews on tap. These are always changing. What is there now will probably be totally different then what is on tap when you come down. If you like beer, this is a place is a must visit.
Enjoy your trip.