Some people use a thumb tack through the screen and into a wooden branch. I do this but use small stainless screws instead of tacks. I also pre drill small holes in the branch so it doesn't split.
I have everything suspended from my live plants. I also bought some wood dowels to push into the pots and use as additional support to hang vines from but didn't wind up needing them.
The tack method is also good I just didn't like the idea of tacks or nails showing on the outside. But that is purely an aesthetic choice.
Bent paper clips or twist ties outside of the screen works well. I also use sticks and plants as my main support and have used black yarn to tie vines to sticks to sturdy it up a bit. My chams have never even noticed it.
We use tie wire from a spool made for gardening. It fits through the screen without damage and we wrap it around the end of the bendy fake vines from Fluckers, sold at Petsmart.
good idea in the fishing line. i use a very thin gauge wire to just tie stuff up. it can damage the screen a little of you tighten it too much. but im not worried about asthetics. they live in the basement.
just got really small zip ties from the hardware store and you have to be careful like Deku said, when you put it through you have to put it back through again but it has to be close to where you initially put it through or else the screening will break. or fishing line is what my other buddy just used. but takes more time