Am I allowed to wear my RSP pt uniform to a civilian gym?

Recruiting came up with the idea and got funding approved for it because of the branding it provides, and the opportunity to strike up a conversation about who the National Guard is, and the unique mission it provides. Good luck on your new adventure, I am sure you have already been briefed on the Stripes for Buddies promotion program. The best thing to do to make life easy at basic is show up in the best physical condition possible and being able to pass every pt test at the full level from the very start. It makes your Drill Sergeant's and your life much easier. It is not an official Army Uniform, so the wear and use rules are different.

A fair bit of misinformation here. Short answer is yes you can but check with your recruiter to make doubly sure.

AR 670-1 is the guiding principle document here and Chapter 10 section 3 of the document dated May 2017 states:

10–3. Occasions for wear
a. All-purpose wear. The physical fitness uniform is authorized for wear on and off duty, on and off the installation, unless restricted by the commander. Soldiers may wear all or part of the physical fitness uniform with civilian attire off the installation, unless restricted by the commander. The physical fitness uniform is not intended for wear as an all- purpose uniform when other uniforms are more appropriate.
b. Approved wear. The physical fitness uniform is prescribed for year-round wear for all Soldiers, unless otherwise directed by the commander. Soldiers may wear the physical fitness uniform off-post unless prohibited by the commander.
c. Restrictions on wear.
(1) Soldiers may not wear the physical fitness uniform for commercial travel.
(2) Personnel may not wear the physical fitness uniform in off-post establishments, unless for purchase of essential items (for example, gas).
(3) The physical fitness uniform is not considered appropriate for social or official functions off the installation, such as memorial services, funerals, weddings, inaugurals, patriotic ceremonies, and similar functions.
(4) Commanders may further restrict wear of the physical fitness uniform per paragraph 2–5c.
(5) The physical fitness uniform is not appropriate for parades, reviews, and ceremonies.

Note the restrictions. These are what is official, UNLESS your commanders policy states otherwise.

You are not in the service yet so only your recruiter or trainer can tell you that. Good luck.