Where exactly will you be staying, near Denver itself? Centennial is
probably the closest to Downtown Denver, and definitely closer if you're on
the south side of Denver. If on the east side there's Aurora Airpark and
Front Range Airport. Then the others that have been mentioned, Jeffco,
Longmont, Boulder. There's also Erie (Specialty Aviation I believe it's
called), worth a look. Personally I try to pick the nearest airport to base
(or rent) a plane, defeats the purpose having to drive over an hour to get
to your (an) airplane.
But I know people who do, I don't get it. Dont bother with clubs,
especially with your situation. Maybe they'd waive the club fee if you take
the CFI with you?
If you're from here you probably know the inherent hazards, if any, at each
airport. BJC-winds, APA-traffic and inept controllers, Boulder-restrictive
pattern, possibly nasty winds, people/airport manager watching and waiting
to turn you in ,and glider activity, Erie-airpark residents have "our
airport, get out" attitude, FTG-very very close to Denver Intl,
Longmont-skydivers (I dont mind them, but you need to know they're there)

Chris