So on my longbow has a piece of I think leather. It looks worn out. What's the best material to put down for the rest? I see a lot of furs. I included a pic of mine

If the bow shoots well, and puts arrows where you want them, I would be very careful about replacing that rest material with something that doesn't have the same thickness.

I like calf hair or soft side of Velcro

I too like the calf hair.

If you are the one who put that wear on the rest and adjacent shelf finish, I'd suggest looking at your tuning.
Something is hitting the shelf pretty dang hard and that would be an indication of a poorly spined arrow.

That bow looks like its cut way past center so it might benifit from a thicker side plate and a tune up.

Yup, tune up and industrial velcro, that is all I use. It too will wear though if thr Bow is not tuned well! SHAWN

I like Bear hair.
Two little strips.
One on the shelf and one on the riser, making two contact points.
I take small scissors and trim the inside edge to an angle
The inside corner of the shelf had nothing for the feather to contact.
This makes the arrow lay into a spot that is less likely to "bump" out.
Also my bow shoots the best with this set up.
The only complaint is the Bear hair adhesie isnt rain proof. So I often take some extra rubber cement to help it.

I always liked either bear hair or Velcro like mentioned. However on the recurve I am shooting now I had trouble shooting off the shelf so I have an old stick on rest on it.

x2 on velcro!

I use the furniture slide material - whatever the call that stuff. It's dirt cheap at the big boxes. It's like soft side of Velcro but has a slicker feel to it and it comes in a sort of buckskin color that I like better.

I prefer calf hair, but velcro works just fine too.

Velcro.

Velcro
Easy to obtain  and very cheap.

I prefer velcro or calf hair but thoseold stick on rubber bear rests work very good on most bows too. I use feather rests on all my bows now. I love those things.They do raise the arrow up higher off the shelf than velcro but they sure are a forgiving rest.

I feel sorry for you Americans, in that seal skin is no longer an option for you.  We can still use it here in Canada.  It is with out a doubt, the toughest, longest lasting, weather resistant and quietest material ever!