So your getting into origami, now you need to take your game to the next level. Try using what the pros use and get some origami paper. It is already square so there's no need to cut your paper, it is just the right weight making those tricky folds much easier, and you can get it in any color you like.
What I like to do is buy a nice 500 sheet pack. With all the colors, that'll tide you over for awhile. Also, check out the high quality origami paper from Japan, these guys mean business. The Japan paper is all handmade authentic origami paper with great designs.