Yeah, been in the industry for a few decades. Started in it back when everything was on coax or twin-ax, only a few terminals spread around an office. Little bit different now...
Good snips, punch down too, VOM, and basic hand tools are the minimum. Understanding basic trouble shooting skills is a definite need, unfortunately that is in short supply.
A fish tape can be handy, what would probably be used more though are what's known as Glow Sticks, fiberglass rods that screw together. Harbor Freight has a cheap set, but most HD, Lowes, supply houses have sets too. Ladders typically (although not always) would be considered a tool supplied by the contractor. Really depends on the type of environments and areas you end up working in, but 6' is rare for us anymore, all 8' or 10'.
The general area you work in can really effect the types of facilities you work in too. Major metro areas, you'll probably see more Enterprise type office space, cube farms with a lot of UTP cable. More rural areas, probably more of a mix of industrial and outside plant cabling. One thing about this industry though, if you're in it long enough, you will be in every type of building, facility you can imagine. Nice pristine new buildings to run down shacks that should have been knocked down years ago. Military installations where they literately blind fold you as you pass through areas to warehouses where the building dimensions are measured in miles.
Of all the trades, I've always found electricians to the most widely versed in the trades, they have to be pipe fitters, carpenters, sheet metal guys, etc. The LV guys should be able to do most of those things too, albeit probably not as commonly, but having knowledge of it sure helps. Understand the other disciplines, know what the HVAC guys are doing, you have to work around their ducts and VAVs. Know what the electricians are doing, you have to work around and through their conduits and panels. If you get into Data Centers and large MDF/IDF equipment rooms, they'll be looking to you for answers. Learn about the equipment you are hooking to. The comm guys, more so than any other trade, become the middle man between the IT geeks and the construction world. The better you can interpret things for both sides, the more valuable you become to all.
JR