Rule 1: Don't try to go get a 21oz playing cue, even if that's what you gravitate toward for house cues.
Thas being said,
Assuming $100 or so is in well within your entertainment budget, you can pick up a McDermott Lucky, Players, Viking Valhalla etc.. type cue. They're good enough and will last a while. 99% of the benefit of a owning a cue is just eliminating the variability that comes with house cues. The remaining 1% is getting a cue that meets your remaining personal preferences, once you know what they are.
You can use your league time to try other people's cues, and start honing in on what your personal preferences actually are. Try a few different shaft diameters, in particular. If, after some time you stick with the sport, replenish your entertainment budget, and decide to upgrade, you can then also just throw a hard tip on the first cue you bought, and presto! You now have a playing cue and a breaking cue.