For low tension connections:

If you need to hold something on, but it doesn't have to hold a lot of weight, use velcro. It's easy to attach to most materials (thermal glue, silicone glue or sewing) and it allows easy adjustment of size in case you have to wear different things under that part of the costume depending on the weather.

For medium tension connections:

Plastic snap buckles and nylon webbing are great for adjusting the size of a connection and then clicking it in place. These can support more weight than velcro with less chance of coming undone but take a little longer to adjust the length if you need it.

For high tension connections:

If you've got to support a lot of weight but still need a connection you can undo - you can use very wide velcro patches if you have a lot of area, several buckles if you have less area, and a nut and bolt if you have very little area.

For permanent connections:

Any two materials can be joined in an number of ways. For most situations, silicone glues can be very effective as they can hold a lot of weight and still are a little flexible. For more rigid connections, epoxy glue can be used.