You might want to consider a more dedicated data acquisition setup like those mentioned above but I'm not sure how you would get started for under your $250 price point without putting in some time in developing input circuits and figuring out how to process the results. Ultimately a low end USB data acquisition system will be much more flexible since it will provide you with at least 4 input channels (typically +/- 10 V) vs. the one to two channels on a DMM. If you are interested I'll dig around some info I have a work.
Sticking with the multimeter approach you should be able to find a capable setup for under your price point. I don't have any direct experience with any of the brands in this price range but I think Greenlee would be a good place to start since they offer calibrated units in addition to standard units. It looks like the DM-510 or 210 would meet your needs and leave you enough to buy the USB adapter kit.
There are a lot of other brands out there, including some found on the multimeter warehouse links others have referenced. After scanning through these and checking out some other web sources it looks to me like this is a good example of you get what you pay for. Several of these meters look like (and the manuals read very similar to) meters offered by more well known (or maybe just more expensive) brands. Reviews for these meters indicate that when they work, they work well enough but there seem to be occasional quality or performance issues.
Best advice is stick to brands/stores you are comfortable with or those that are recommended by people you know and trust. Realistically that means my recommendation above is suspect since I have never used GreenLee products
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I have no affiliation with any of these brands beyond being a regular user calibrated meters and equipment used in testing critical systems.
Good luck,
-Eric