Yes, it's doable with over-the-counter products. You need a meter and lancet device, and then for each test, a single-use lancet insert (tiny needle) and test strip.
These are usually bundled together in a pharmacy for $25-$75. However, if you look or ask the pharmacist, there's often a 100% money back refund right in the store.
Additional test strips are $0.75-$1.75/strip in packs of 25, 50, or 100. While that sounds expensive (and for diabetics, is), for the purposes we're talking about, one could probably only test after high-carb meals and a few days of fasting readings. The included 25-50 test strips may yield enough data to act on, and if not, one more 25-50 pack probably would.
https://www.diabetesselfmanagement.com/blog/what-is-a-normal... has more about when to test (basically, 1 and 2 hours after the start of a meal) and what results to expect. Feel free to ask the pharmacist to show you how to use it and interpret the results.
Important note: this feels nothing like a blood draw. It's barely noticeable at all and not anything I'd call painful. There's a depth adjustment on the lancet, so it's the tiniest imaginable prick on the side of a finger. It yields less than a drop of blood, then immediately closes, ie, no bandage or bleeding.
I did this myself as an extension of quantified self in 2012 (https://twitter.com/troyd/status/281931325471068161), just because I wanted to see the readings and the process. I did it again this year to see whether the testing process had improved (no) or my results had changed (no). Other than some puzzled pharmacists who aren't used to people buying a meter for cash, there's really nothing to it. It'll also give a tiny peek at a process that millions of people do all the time.
It's fascinating to see how one's body responds, particularly to meals that aren't obviously unhealthy (and depending on the person, may not be). Some examples: sandwiches (especially on big hoagie/grinder-style refined bread), burritos (tortilla+rice+beans), any meals with 2 significant sources of carbs (sandwich with fries or potato, breakfast with toast and hash browns, etc), and pasta. These aren't inherently unhealthy, but they all significantly raise blood glucose, and the only way to see how much is to test. More on "glycemic load": https://www.google.com/search?q=glycemic+load