So for the initial kit, it's designed to connect the panels in parallel (there should have been a Y-splitter in the pack) to the controller, 14AWG wire would be fine. The inverter is going to be wanting to draw almost 50a of power from the battery which would be 6AWG wire, although the battery is going to be maxxed out on its BMS to do that.
From the inverter to a breaker panel (which is overkill for that small an inverter) you can use 14AWG wire because it's only about 5a on the 120v side. You can use a power strip plugged into the inverter (since I doubt it has terminals) or cut up an extension cord to go from the plug on the inverter to the breaker panel. When you connect to the lugs in the breaker panel run a jumper wire between the two lugs and you'll have 120v on both sides, but no 220v so don't try. Grounding will go from the ground lug in the breaker panel to a copper rod in the dirt outside and there'll be a screw to make the Neutral-Ground bond inside there. I used a basic
Eaton 2-slot panel and a
couple tandem breakers for my shed when I hooked everything up, it was just right for the small space.
As to the tools you want to use, a skill saw is going to want about 1000w to run, maybe more depending on the size of the saw. The air compressor is going to want pretty much the same. Either way the startup surge is going to be really high and your inverter is probably going to trip out even trying to start those.
The kit you've got can easily recharge batteries on cordless tools, and it'll run lights for a while, but directly powering tools is going to be difficult at best and impossible at worst.
Once you've got the bug and start thinking about upgrading the system, you'll want to start replacing parts in (IMHO) order:
MPPT Controller - 40a or more (Pro tip, if it has USB ports on it, it's really a PWM in disguise and ripping you off)
LFP battery - 100Ah or higher
1200-1500w Pure Sine inverter
Shunt
More panels.