Looking for suggestions on a nice scope for my Rem .22 Speedmaster. Have had the gun 30+ years, it had been back-and-forth between family members and has finally reappeared in my gun safe but without the scope. I remember putting a cheap $19 Tasco scope on it when I received the gun from my parents back in '83. This go around would like to step it up a notch.

Any suggestions?

Weaver V16. This is an excellent scope for a rimfire rifle, works well for target and hunting.
Weaver V-16

If you want to step up even father, go with a Sightron Sii Big Sky 6-24. They go for about $575.

Resist the urge to get a $100 scope (which I'm sure numerous here will recommend). You will be happier in the long run with a quality scope.

I have rifles with scopes that cost more than the rifle and I never go to the range and wish I had a cheaper scope, but I have in the past bought cheap scopes and wished I would have bought a better scope to begin with.

Buy once cry once buy twice cry twice. Choose wisely.

nikon is always a good choice.
prostaff with bdc has treated me nicely.

Neat little rifle. I'd go with a smaller scope, something with a 28mm to 32mm objective.
The Nikon 4X32 should work, even if it does only cost about $100.
Or a Weaver 4X or 2-7X rimfire model.
Or a Leupold 4X or 2-7X rimfire model.

Thanks everyone for the time & suggestions.

Would you know what scope rings i'd need for the Remington 522 & the Nikon 4x32?

You need 1" rings for a 22 with a grooved receiver. Wherever you buy the scope should sell rings. There are lots of different brand names. The cheapest Chinese rings should probably be avoided.

As stated, buy once and you will save time and money in the long run when buying a scope. A good brand name scope that can be changed from 3X to9X or more is what would be best IMO and can be moved to other rifles as needed. I have a 4X14X44 on my .22's and use them at the range and for hunting , and they fit the use just fine. Cost more to buy but I have never had to buy another scope for any reason. JMO
Oldman570