One of my principles is that an employer has to pay for all equipment I need for my work. PC, laptop, phone, measurement instruments, software, tools, ... If it is for the job then The Man has to pay. That is part of being an employer. If I want to pay for my own stuff I would be self-employed. So I am surprised that you want to get a meter and network tester on your own.
(I make an exception for pens and pencils, as my employer's seem to magically end up in my pockets and at home. Therefore I occasionally bring some from home, which might anyway belong to my employer).
But OK, regarding network testers, just like multimeters, there is everything available. From simple cable continuity testers, to complex protocol analyzers with all sorts of measurement and analysis features. Beyond continuity testing things get expensive, fast.
You need to figure out what the most common faults are in your job and then look for a network tester with the ability to identify those (and a few more).
For protocol analysis stuff a combination of a laptop, Wireshark, and often a small switch capable of port monitoring is also common. However, that setup is bulky in the field, any you don't really see what is going on on the wire, just what the laptop's (or switch's) network interface make out of the signal level on the network cable. .g. you don't see bit error rates.