I'll be transferring my few fish to a larger new tank (10 gal to 40 gal) this weekend. Not sure if I will start from scratch as far as cycling is concerned.

I may have the chance to obtain some gravel or small stones from a long time established (and healthy) fresh water tank.

Are there any benefits to this? would this speed the cycle and make it better for my fish sooner? what are the downsides?

My new aquarium came with safe start (a bottle of beneficial bacteria) - is this worth adding and is there a benefit?

Yes, it will help the cycle and speed it up. Bacteria attaches to mostly the gravel and your filter. How much it will help is hard to say, but it should shorten the time required to complete the cycle. You can put the used gravel in a filter bag or if you don't have that, an old stocking will work. Just remember to keep it wet with tank water. The bacteria will die if it dries out. You'll also need some type of ammonia source in there - either fish or dosed ammonia.

Safestart is up to you. It doesn't hurt.