For rifle ranges in Tacoma (and the immediately surrounding bedroom communities):
It has one public outdoor range. Tacoma Sportsmens (200 yd public range).
It has two membership outdoor ranges. Paul Bunyan & Tacoma Rifle & Revolver.
The greater surrounding area for public rifle ranges have Renton Fish & Game Club (200yd), Black Diamond Gun Club (open to public on specific days, unk distances), Cascade Shooting Facilities (formerly Cascade Rifle And Pistol Club - looks open to public on every other Saturday, unk distances.), Evergreen Sportsmans Club in Littlerock (unk distances), Kitsap Rifle & Revolver (unk distances)...
There are plenty of places to shoot "in the hills" if you'd rather plink in a dirt pit, etc...
There are several members-only clubs in addition to the above that, if you find a member, can get you in for some guest shooting... (Capital City Gun Club, Upper Nisqually Sportsmans Club (which may have some public-open days.... it keeps changing)
As with any range, call ahead. Make sure they're open, open to the public and have an RSO on duty where needed - nothing more frustrating than packing up all the gear, making the trip and finding out "Todd" didn't show and without an RSO, no shooting is allowed at Range X.