Here are all of the ways I can think of to stick things to walls and together, in order of awesomeness:
Thumb tacks: Endlessly reuseable. Go for the old school all metal ones with the round flat head so that there is no plastic problem when they break.
Home made glue: Flour and water is a really strong, non-toxic glue. Drys white so be careful if using it for wrapping presents.
Paper tape: Masking tape and brown paper packaging tape will both breakdown in your compost. Available from most stationers.
Brown string: Instead of plastic or nylon string get a ball of real brown string. Its more fun.
Blu tack: Made of synthetic rubber, polymers and fossil oil, it probably won’t be breaking down anytime soon but it is remarkably reusable and multipurpose. Beats cellotape!
Removable plastic hooks for artwork: Popular because they don’t leave a hole in the wall, especially if you are renting. Plastic 😦 …but durable 🙂 …and not easily substituted. But if you can, use a nail instead. Its not big deal to whip round the house when you leave with a bit of plaster and paint (if you have some, trickier if you don’t!).
Liquid glue purchased in large containers: “regular glue” like PVA etc are just plastic in yet another form. Wont break down, can’t be composted. However, you can still do better than a glue-stick. Buy the largest container you can, like a one or two litre, and refill your smaller container with it.
Sellotape / scotch tape, packaging tape and duct tape: Any type of plastic tape is single use, and not designed to last. Duct tape is a classic, we all use that to fix everything but its a quick fix. If you can tolerate it, take a step back, ask yourself how can I solve this without adding more plastic to the environment.
Glue sticks: bottom of the barrel here folks.
Cable ties: Of the devil! We hadn’t even heard of them in the 2000s but now we can’t seem to live without them. Find your nearest scout and ask her how to tie a knot.