Update: 
Alright, I feel very stupid about not thinking about trying this before, but today I tried to play with my amp in my room and me standing in the hallway and record it. After listening to it, I could barely hear the clicks. Because of a cranky grandma living with me, I always played either very quietly or using headphones and I always heard clicking, but if I turn the volume up my problem almost disappears and those very faint clacks I'll put down to my inexperience. I really feel quite dumb for all you nice people trying to help when the problem was just in me not exploring it well enough.
I was so surprised by the quick and numerous responses I recieved, TB really seems to be a great site! I can't answer all of them, but thank you very very much for all your advice, it is very useful. I'll try out your tips about technique and I'll get my bass professionally set up as soon as I can, it can't hurt.
Again, thanks so much for all the answers and I'll be sure to thoroughly check out whether I do really have a problem before I post next time.
Rock on!

Alright, I feel very stupid about not thinking about trying this before, but today I tried to play with my amp in my room and me standing in the hallway and record it. After listening to it, I could barely hear the clicks. Because of a cranky grandma living with me, I always played either very quietly or using headphones and I always heard clicking, but if I turn the volume up my problem almost disappears and those very faint clacks I'll put down to my inexperience. I really feel quite dumb for all you nice people trying to help when the problem was just in me not exploring it well enough.
I was so surprised by the quick and numerous responses I recieved, TB really seems to be a great site! I can't answer all of them, but thank you very very much for all your advice, it is very useful. I'll try out your tips about technique and I'll get my bass professionally set up as soon as I can, it can't hurt.
Again, thanks so much for all the answers and I'll be sure to thoroughly check out whether I do really have a problem before I post next time.