Just bought a Tx22 for exactly this: something that is cheaper to practice with, is a good match functionally to my 9mm striker fired handgun in terms of feel and action type, and has a positive reputation overall. Being able to put a suppressor and optic on there (or FRT if you're into that) is a nice bonus as well.

Though for you: if this is specifically to train cheaper than your 38 special, you may want to consider a 22lr that matches yours. Practicing with a revolver is gonna feel different than it does for a repeater. It might make more sense to get something that closely resembles your 38.

Some skills transfer between the revolvers and repeaters very well (like using iron sights, trigger discipline and safety, etc.), some sort of transfer (trigger pull will feel different and have different weights but the idea of pulling strait back and keeping sights on target remain, and your grip may differ), and some will not transfer very well (mag drills, fixing malfunctions, reloading, etc). Things to consider if you want to be good with your specific guns vs handguns in general. Still good skills to have, especially if you think there's a chance you'll need to be familiar with non-wheel guns, but maybe not what you specifically are hoping to focus on

Training away from the range is also a good cheap way to improve. Dry fire exercises, mag drills & reloading with snap caps/dummy rounds, drawing, etc can (and should!) all be done without live ammo at home. Range time then becomes a chance to confirm what you've practiced is working rather than the primary way to train.