I'm 21. I've had a bump grow on me penis and go away before. This time it stayed for awhile and another one grew not too far from it on the shaft.

After letting it be for a week or two I decided to pop it but it wouldn't. Than I got a needle and poked it. It let out a whole lot of puss and got smaller but it still there. The second one I also popped and it let out some white stuff similar to puss but its also still there and have both been there now for about 4-5 months. I haven't stopped trying to pop them in the meantime so I guess that would have something to do with them being there still.

I am ready to go to the doctor, but I don't know where.

I mean, do I just walk in to the nearest walk-in clinic and ask to see the doctor? Or is there some sort of specialist I should call and make an appointment with?

A G.P. should be able to answer your concerns and be sure to tell them how you took care of the first bump.All of the history is relevant.
If you need to see a specialist ,you will be referred.
If you have walk-in clinics in your area do get seen.