I am of the opinion that beginners should buy the best quality they can afford, precisely because they do not yet have enough experience to diagnose and work around limitations or defects in cheap stuff. (That doesn’t mean buying top-of-the-line everything, but it does mean avoiding junk, especially among product types where some stuff is really, really bad.)
First off, I’d recommend using an I2C-based environmental sensor instead of the DHT22 which uses a weird protocol that is timing-sensitive (and thus easily disturbed by other code running on the MCU). For example, try the SHT45.
Modules from Adafruit, etc. cost more than no-name ones from AliExpress, but you know it’s genuine and will work properly.
Breadboards are a product where you
absolutely want high quality. I originally bought a number of cheap breadboards (before later investing in some top-of-the-line 3M ones), and compared to high quality breadboards, all of the cheap ones have quite low pin retention force (so it’s much easier for components with thin wires to come out unintentionally). Some have badly shaped contacts that snag on the component leads, making it hard to insert components and wires. And some have contacts so bad that they have many ohms (!) of contact resistance, which can affect some circuits (especially analog), but also means that if you run significant current through them, they can heat up.
But the worst is if you have a poor connection or a component pulls out unnoticed. Then, when the circuit doesn’t work, you don’t know why: is the circuit design wrong? Or is it just a bad connection? You will avoid a lot of frustration by using good breadboards.
In the end, cheap breadboards are just not reliable enough. Some will be OK, others unusable, so if you do must cheap, order more than you need, and choose several types from different vendors, in the hopes that some will be good, but don’t be surprised if some are essentially trash. (At least the trash ones do have one purpose: I use them as holders for soldering, like when soldering pin headers onto a module.) There have been many previous threads about breadboards, and people have recommended various brands that are good, but not as crazy expensive as the 3M breadboards now are. I recommend some mid-priced brand like Global Specialties.
At this stage, buy jumpers. You can get high-quality jumpers from Schmartboard on Digikey, etc. Others here have reported good results with
silicone DuPont jumpers from AliExpress. The cheap standard AliExpress ones are usually OK, but the female contacts wear out quickly, and the wire can become brittle with time, so you just have to consider them as semi-disposable.
I would not recommend trying to make your own DuPont jumpers yet, because DuPont contacts are actually unusually fussy contacts that do not crimp properly with nearly all cheap “DuPont” tools, no matter what the tool vendors say. I have a
big thread specifically on this topic if you are interested.
There is one exception: if you have Lidl where you live, and you see Lidl’s ~€20 DuPont & XH crimp set that they sell once a year, buy it: it is the ONLY cheap DuPont crimper I have tested that actually produces decent crimps. (They’re not as perfect as the crimps from the expensive tools, but the results from the Lidl tool are actually
good, i.e. they do not pull out and do not mangle the contact.)
What you can make yourself are the non-DuPont wire jumpers for the breadboard itself: just get 0.5-0.6mm diameter (0.2-0.34mm², 24-22 AWG) solid wire and strip 6-8mm from each end.
Power supply: well, I certainly do consider a lab power supply to be essential to electronics, and current limiting is great for protecting against beginner mistakes (and non-beginner mistakes!

). Luckily, you can get great bench supplies affordably. I highly recommend the ones from Korad, particularly the ones that have preset memory and an output on/off button, like the KA3005 series. (The KD3005 doesn’t have those.) I own the KA3005P and it is my primary power supply at home.
With that said, for pure MCU projects like this, you’ll be connecting USB anyway to program it, and in many cases you can just power it from that. An inexpensive, simple precaution is to use a powered USB hub (i.e. one that comes with its own power supply) in between, so that if anything goes horribly wrong, at worst you’ve fried the hub, and not your computer’s USB port.
There are many threads about cheap multimeters. I think $20-50 is a great price range in terms of bang-for-your-buck. For example, the Aneng/Zotek meters. FYI, you might be able to speed up your current meter’s continuity test by very thoroughly cleaning your probe tips with alcohol. (Or getting better probes. And still, clean them.)