FWIW ~2 months has sometimes been time it's taken me to find the used car deals I was looking for. Once I've been immensely lucky and stumbled on an incredible deal quickly. More often it's in between.
I'm not an expert, but my tips: figure out your budget (mine is usually $2-4k). Then cast a broad net for cars that have less than 120k miles. Many cars made in the last 20 years have upwards of 150-200k in them. So starting here is going to give you potential for ~50k at the cost of a down payment. It can help if you have a few models in mind that you know tend to have particularly long lives or tend to be undervalued (my favorite for a while was the Geo Prizm, which is a rebadged and therefore discounted Corolla, I've also had two Oldsmobile Cutlass Cieras that went into the 200k range... but these were both discontinued 15-20 years ago and good ones are getting rare). Do your own filtering on the health of the car. I'm not even an amateur auto mechanic but I have a few heuristics: is the temperature stable over a range of driving conditions from idling to hill loads or freeway? What kind of oil leaks do I see? Also I like visible brake & gas pedal wear as a proxy for general vehicle wear. If a car doesn't have any obvious problems on these points, fits my cost/miles profile, and I like it, then I take it to a professional mechanic for a pre-purchase inspection and follow their advice. If the mechanic notes some needed non-powertrain repairs that fit inside the budget, I'll usually buy, perhaps after some negotiation.
There's a downside to this approach: it can be time-intensive. If you're well-compensated with your job (or you're doing a startup, or you're a new parent, or you have some other time-intensive thing going on in your life), your time may more valuable than the economizing this approach can yield. It goes faster if you let out the parameters a bit (bump up the budget and take down the mileage). In most sufficiently large cities there's probably a If you want more tips, Google Greg Macke, who seems to have some sensible advice about evaluating cars and has some recommended/lemon lists.