Dwillems26

I have 2 bikes, 1 with a cassette, the other a freewheel. On both bikes, if I grasp the big ring I can wiggle them, the whole assembly slightly wiggles. I know about freewheel wobble and this isn't that.

I'm assuming I know the most likely cause, but I want to heat some experience and advice.

The cassette probably just needs tightened to the hub, usually a 10mm hex key.

The freewheel it looks like the outer assembly is wiggling, but the inner casing seems to stay solid. So I'm assuming it's the bearing race/retainer that needs tightened up. Either it's loose or I need to remove a shim under the race. ?

I was going to pull it apart last night, but realized I didn't have the right size cone wrench so I didn't want to mess with it yet.

tFUnK

It's more likely that your hub cones need tightening.

dabac

The cassette mounts to the freehub BODY, which in turns mounts to the hub. What you suggests will tighten the f/h body to the hub while doing nothing for how tight the cassette is mounted to the f/h body.
If this refers to first disassembling the f/w, then removing a shim would be my first guess.
While ”things coming loose” is always a possibility, keep in mind that the only load a freewheel or freehub carries as it’s coasting is the chain tension. And particularly for derailer gear bikes, that ain’t much. A little slop doesn’t hurt anything and speeds up production.
If you insist on doing something about it, removing a shim may well be what the cassette wheel ”needs” too. For Shimano f/h, there usually is one or two under the DS wheel bearing cup.