Rugs and scuff pads.

Can anyone tell me if there is any “Type” of rug, beside rubber backed rugs, that don’t side all over the place as you wall on them?

Since we’ve installed our hardwood floor, the rugs wonder and slide around as you walk on them. Is this something that I just have to live with?

I’m also having problems with furniture sliding around on its felt pads as you sit. Is there any way around this?

Sounds like you have Sweedish finish or even tung oil finished floors ...
A carpet/flooring store should carry a few brands of rug pads that are slightly rubberized. Don't go cheap. Some less expensive brands will break down (especially in a sunny area) and stick to the floor after a while (year or more) leaving a messy lean up.

Also there are rubberized black soft furniture leg/feet sticky back or nail on backed pads at some of these floor stores as well.

Lowes flooring dep't has rug pads too. And thier hardware dep't has a brand of soft black rubber furniture pads.

Good Luck!