My first post on a thread here so please be gentle. I'm an electrical engineer and have worked on many instruments and power systems over the years. Would just like to add several thoughts and cast my vote for several others.
For the OP, the best way to learn what instruments can do for you is to learn basic ac and dc. Learn the difference between voltage, current and resistance. When troubleshooting an ignition, charger, kill circuit, measurement or reporting circuit, resistance measurements alone will frequently lead you to a fix. You may need to find some voltage that fail before you can apply the resistance testing of wire harnesses, switches (on and off states), ground connections, etc.
On instruments, Fluke is a good choice. Fluke, Keysight, Tektronix are some of your top electronic suppliers. Many of the other names may be ok or may be garbage. It's best to ask around with others you know. The top companies really know how to measure stuff. And they know how to measure it when the environment is poor (temperature, humidity) or can effect results (close to RF transmitters, emergency communications, even cell phones.
On probes, most of the quality meters come with decent probes. I have a number of the Probemaster hard gold plated probes and they really are great but for normal troubleshooting, just keep your probe tips clean. The benefit of gold can be lost in the noise on a piece of outdoor machinery where on a bench on sensitive circuits, the gold really makes a measurement attain its best accuracy.
On batteries, I've lost many pieces of expensive and in some cases sentimental equipment due to the bleeping alkalines. For my DMMs and any valuable equipment, I now use only Lithium primary (non rechargeable) batteries. Mostly AA but also some 9V and other sizes.
Also in the spirit of the OP's questions (I hope), check out various electronics forums on line. Most will have a beginner spot but in general, you will be treated with respect regardless of where you go. And the knowledge out there is tremendous. Even electronic nerd or geek types have grass, brush and trees to cut