Last year I bought myself a reasonably price Aoyue Int701A++ "repairing system", which include a variable temperature soldering iron (with integrated smoke/fume sucker) and desoldering pump (good for taking stuff off boards). It is mid-range in terms of quality.
Iron is good. The desoldering pump is OK but seems to always need cleaning. However my solder sucker/solderpult (hand/spring-powered) still sucks hard and rarely lets me down. You should also grab some "desoldering wick" or "braid" as that can help in situations where the solderpult can't. I still prefer the solderpult or desoldering pump where possible, but sometimes using the wick is the trick.
Ordinary solder will do for most jobs, there is no special solder for monitor chassis. Typically you want the stuff with lead in it as it melts at lower temperature. For general purpose soldering (which most of it is) get solder with rosin/flux core 0.7mm diameter with a 60/40 or 63/37 ratio mix (tin/lead). I also have some really thing 0.25mm solder that I use for touching up really fine and tiny stuff.
If you want to do SMD work then you need a decent set of tweezers and solder paste. Mechanic brand is good, I use it from a small pot and also from a syringe. You'll also want some solder flux which helps the solder melt at higher temperature, handy for difficult spots. Flux also comes in syringe packs.
For SMD work it really helps if you use a hot air system. I use a "XXX 858D" which is a commonly cloned model cheaply available online. Mine actually says "JCD 858D" but there are many variations. A bit of getting used to, but it really makes life a lot easier for doing that tiny stuff.
You'll also need a combined magnifier/lamp on an arm, to help you see your work. You may also need magnifying glasses (they come with interchangeable lenses) and/or a secondary lens (use it in combination with your magnifying lamp to get greater magnification, just like what happens in a telescope/binoculars/microscope).
Get a solder scouring pad, like a steel wool pot cleaner in a pot, for cleaning and re-tinning your solder iron tips.
Roll of masking tape for lots n lots of things.
Some kind of stand or clamps or vise to hold your electronic boards while you work.
Tiny screwdrivers, pliers, wire cutters, old rag, toothpicks (really essential for SMD work, tidying up solder paste before blasting with hot air)
Get a digital multimeter if you don't already have one. I have many cheap DMMs ($10-$50) which are fine but the Fluke 17B+ (around $150-$200) is like wiping your donkey with silk by comparison (you have to say that like the Mandelorian from The Matrix movies). Grab some alligator clips that you can slip onto the ends of the probes, makes life easier.
Isopropyl alcohol for general cleaning up, but it kinda just moves old flux around. So, clean soapy water and toothbrush for really getting stuff clean once the job is finished (or before you do the job if chassis is really filthy). After washing with soapy, rinse with clean water. After gently padding stuff to semi-dry, use some isopropyl to help remaining water dry off quickly and leave less marks.