There is nothing wrong with saying "I go weight training every night'.
You can nearly always say "I go *******' if the ********* ends with 'ing' and the word relates to an active physical activity. There are lots of exceptions to this though!
eg. you can say:
I go running, I go sailing, I go singing, I go playing, I go volleyballing, I go riding, I go driving, I go golfing....
but you would not say:
I go lighting, I go winning, I go growing, I go painting, I go reading, I go making, I go looking, I go gardening.
I know that this doesn't help much!
If in doubt, you can always replace the 'ing' and say 'I go to play', or 'I go to garden' or 'I go to read'....but this would often not be natural English. This is one of those bits of English that you just have to learn which usage is usual and which is not usual (I never say right or wrong).