Electrical Circuit Diagnostic & Building/Reworking
Tools That I Use & My Experiences
There are plenty of tools involved in circuit building, testing, diagnosing, repairing, etc. At the head of the pack is probably the soldering station and the digital multimeter (DMM). And when I say "soldering station", I mean soldering iron/rework station/hot air gun/solder sucker/3rd hand,hot tweezers... basically what you would find at any number of soldering benches from a once-a-year repairman to a high-end custom circuit builder. And by DMM, I mean anything from a $3 garage sale multimeter to a clampmeter to a forkmeter, to an oscilloscope, along with any and all add-ons and accessories (different leads and probes jump to mind first). You can get a soldering iron for under $10 and a DMM for free (with coupon at everyone's favorite super cheap local tool retailer) if you really need but have the kind of budget I'm working with.

Something super helpful that I learned is that most meters have the ability to perform exceedingly well. The cheap ones are almost always held back by garbage leads/probes. But you can get a fantastic pair of aftermarket probes for under $20, delivered in 3 days, from ProbeMaster.com. With the added bonus of supporting American manufacturing in a time that they're very few and far between. In fact, I'd be surprised to hear of one other lead/probe manufacturer that is. But I added one of their pairs of the 8000 series leads with retractable-shielded straight banana plugs to a cheap generic clampmeter and the difference was IMMEDIATELY noticeable. It brought the performance up to the level of my friend's Fluke that he paid 5+ times the price for. And there are a number of comparison videos on YouTube to back this statement up. Eventually I hope to add a 9100 series set, because the different tips would simplify more than a few of my projects, which would also allow me to move the 8000s over to my higher risk DMM allowing me to get decent quality readings in those situations too. I've also seen a new DMM that looks absolutely AMAZING for my uses. They're calling it a graphical DMM, and it's kinda the missing link between DMM and oscilloscope. Ideally, I'll be able to upgrade my setup in the near future to one of these graphical meters with the 9100 lead set. If you're in the market for a meter now though, I HIGHLY suggest you take a look at graphical multimeters (I don't have any direct experience, but the latest Mustool has a crisp display, can plot graphs - sine/square/triangle/etc., and even save data... for less than many regular DMMs) and aftermarket probes (ProbeMaster are the only ones I have experience with, and it's been a fantastic experience. Well worth the price. Lightning-fast delivery. Great customer service. And support US business). I really doubt that you'll be anything other than EXTREMELY satisfied. As I'm damn near obsessed with getting my hands on them.

As far as soldering, I've only even been soldering for about a year. And started my journey into soldering with a generic 853D SMD Rework Station. It cost just shy of $100 and probably would have been great for someone that only needs to solder fairly rarely. But mine didn't even last me the year. First I blew out the adjustable DC power supply using it for electroplating. And then the soldering iron died after owning the station for maybe 8 months. I replaced the iron, but it didn't help. So I just canned the whole station. Since it was my only soldering iron, I needed to get a new one ASAP. I liked the thought of a portable soldering iron and then discovered the TS100. It sounded incredible. And I could even get it with the conical chisel tip, which of the roughly 20 soldering iron tips I had for my 853D, I used the conical chisel maybe 90% of the time. But the $80 price tag was a bit steep - I'm sure you guys that do a lot of soldering are rolling your eyes at my ignorance, but that's just my reality. While I have done a LOT of soldering in the time since I started, that time has only been about a year. I also came across an open-source project in which you can build a universal soldering station capable of controlling many hundreds of soldering irons called the "UniSolder" that was also incredibly appealing. A plug-n-play controller for a generic Chinese iron or high-end JBC and everything in between sounded like something that would be impossible for me to go wrong with. But it required assembly (AKA a soldering iron, which I no longer had), so I decided to just bite the bullet and get the TS100. I can't speak for the stock firmware, as the second I took it out of the box, I flashed it with the latest open-source firmware, and that has been incredible. I never would have guessed that 2 soldering irons could be so different - and that the better of the 2 was the one I could throw in my pocket, and not the one with a cubic foot control box. It's such an improvement that I still haven't gotten around to buying/building the UniSolder + JBC Iron. Though I still definitely intend to. I use the TS100 multiple times per day, and it's been so incredibly flawless. I've seen that the next one in that line is almost ready to hit the market - with the working name TS80, it uses a USBC plug instead of the micro USB + DC plug the current TS100 uses. I'm definitely curious about that one. Like, what direction did they go with it - is it more portable than the TS100? Is it more of a stationary iron with increased performance? Maybe the same general application, but cheaper or simplified? I'm going full-blown OCD about it. But if you're in the market, I can't suggest it enough. Not all digitally adjustable soldering irons are created equal (as I ignorantly assumed at the beginning). And I'm a very big fan of open-source products. But it's a fantastic iron. Portable and precise, fast to heat but will automatically cool down if you leave it sitting - so you wont burn your house down if you fall asleep or leave the house with it on. It can be set for right or left handed operation. And as a lefty, I can attest to how well it works in said configuration. It can easily be set for various power supplies from different DC adapters to different battery packs. It's a brilliant design that can be used all over the place. While soldering irons are important, so is the solder itself. If it won't flow and bond properly, your joint will suck. Some people may be fine with simple non-adjustable AC irons, but not me. I need as much help as I can get. Thus far, my favorite solder has been Chip Quik 62/36/2 Tin/Lead/Silver. It melts and flows easy, bonds well, and then the factor that made it so appealing to me in the first place is that silver conducts so much better than tin or lead that the little 2% actually makes a real difference. And I may be beating a dead horse here, but I have very little idea what I'm doing, and I'm using components made for totally different appliances/devices. I'm far from being able to compare with the tools made by professionals with specifically selected components. Therefore, I can't afford to handcuff my circuits if I want tools that I can actually work with.
Diagnostics & Repair Equipment
If you're going to be building or repairing stuff you're going to need an assortment of diagnostics tools. And then there's a ton of other tools that you don't exactly need but will make your life much easier.

Probably at the top of the list will be a digital multimeter (DMM) and/or oscilloscope to be able to see the voltage/current/resistance/etc. in your circuit. If this is all new to you, you're working with a limited budget, and you're not sure that this hobby is for you, I wouldn't even look at oscilloscopes. Something that I was fortunate to learn early on into this hobby is that good test leads are essential for accurate results. There are a handful of brands that make good leads. The top two that I'm familiar with are Pomona and Probe Master. Though I only have personal experience with Probe Master. My main meter is a (relatively) cheap clampmeter from the local hardware store that set me back around $40. Continuity testing had a lot of noise. Thinking it was the meter, I exchanged it but got the same results. So I ordered a pair of 8000 series leads from Probe Master for like $18 delivered. Upon receiving them 3 days later I saw an immediate dramatic improvement. If you're looking for the cheapest possible option, I'd suggest the Probe Master leads coupled with a Harbor Freight meter that you can get free with a coupon. You won't have a million features in the meter, but you'll have all the basics and get pretty accurate readings due to the nice leads with a total cost of $18. And if you decide to get a nicer meter, you can carry the leads over to it as well. Because of the specific needs of my projects, a clampmeter was particularly useful - projects like welders use high currents that would blow a regular DMM. But clampmeters also lack other features that would be useful in other projects. I JUST preordered the updated version of a "graphical multimeter" - which I understand to be a kind of missing link between meters and scopes - for $40 on a Cyber Monday sale. Though I won't receive it until the end of the year unfortunately. I'm hoping by then I'll be able to pick up a 9100 series kit from Probe Master to go with it. I've been very satisfied with my 8000 series leads and the 9100 kit has WAY more applications. But we'll see what my budget permits.