Closest, easily accessible true "sport" routes will be at Camelback Mountain / Echo Canyon Trailhead. The Headwall is a ~15 minute hike uphill from the parking lot and has a handful of decently bolted routes from 5.4 to 5.11. Donamatrix is a fun 2 pitch 5.8 route that leads to the upper plateau where one can find the Praying Monk formation, which has a fun, easy 5.3 route up the east face with lots of exposure and great views (though has some ~20' run-outs between bolts, but on easy ground). Camelback Rock quality is definitely on the crumbly side, but the routes up on mountain-project are fairly well travelled and you'll mostly only be showering small bits of fine gravel down occasionally. not unheard of to pull larger chunks out though, belayer-beware.
Pinnacle Peak and Mcdowell mountains offers some great trad/mixed climbing on granite. Beware any route labelled as "Sport" out here because it was likely established a long time ago and will be not only sandbagged but also likely very run-out / risking ground fall at every single bolt clip. Sven Slab offers a good variety of routes and is close to the tom's thumb trailhead parking lot. Quaker Oats is the classic route out there to get up a ~5.6 slab (sport), does have a 30' run out at top but over easier terrain, then can use the anchors to top rope the 5.7 - 5.8 climbs to the left of it. Girlie-man wall is also easily accessible and offers some better bolted short sport slab routes. Past Girlie-man wall is Sven tower area, Sven tower 3 has some really fun long single and multi-pitch mixed routes, and a few good sport ones too. Spinal tap is a really exciting 5.8 mostly sport (one spot for a cam) that requires you to stem off of a face into a chimney. If your up for a trek, Gardener's wall is a huge, ~250' tall face of un-broken granite with some nice 2 pitch trad routes out there, A lot of people will do Hanging Gardens (5.6 trad 2 pitch), then go down the back-side to Tom's thumb to do another route up there. Tom's thumb is the namesake granite dome with routes from ~100 - 150' with lots of fun trad routes, Treiber's Deception is a nice 5.7 (originally rated 5.5., hence the "deception") that offers some stemming, small crack sections, slab climbing, and off-width in one pitch.
There's other places nearby that offer climbing but I wouldn't pick them over the ones mentioned above. If your willing to travel a little, Queen Creek Canyon is about an hour's drive Southwest to Superior Arizona and has lots and lots of fun sport climbing, popular spots down there are Atlantis and the Pond.
for February climbing, read up on closures for bird-nesting, likely parts of camelback and the mcdowells will be closed (but not all).
Have fun!