Being you have a Tony Kart and LO206 you have a nice map to follow in terms of setup and maintenance. So baselines are available.
I tend to treat chassis and engine separate as engine packages change.
After Last session of a dry day with an LO206:
Clean the chain & let it start to dry
Drain oil - while it is warm ( At least check the oil if you don’t think it needs changing)
On the LO206 you have to flush the carb of fuel to keep it from corroding during storage. This is best done before putting it back on the trailer after the last session of the day. Must of us just drain the fuel, use some carb cleaner and then use WD40. Drain the line back into the tank and reattach back to carb don’t leave it to leak. I also don’t trust Non-Ethanol fuel. So it is just an easy habit to form.
After that I check the clutch end play and pull it. I may just spray it with WD-40 (depending on clutch) to let dirt loosen. Later @ garage I clean it and get it ready to go back on. - What clutch do you have?
Apply Lube to rear axle to help keep from rusting.
That way the engine has every critical thing done that is time sensitive.
Air filter can wait. If you do take it off have a method of blocking the opening. Cleaning, can wait or you can work it in but I’d let the engine cool some and not attack it while it is too hot.
On the LO206 you will want to check your exhaust mounting, head gasket, valves on some interval.
As far a tuning you have gearing & carb.
I like to go over it in sections.
Wheel and tires
I like to check things in order of safety and recurring problems.
2 thing I vowed never to have happen was losing a wheel and brake failure.
Next I added steering catastrophe. Those 3 thing are not just DNF’s but could really get dangerous. I don’t need to add weight, but on a kart with weight added check the weight for security. Losing a weight could be a DQ or really hurt someone. So those are the biggies along with fuel. I wear fire rated gloves and shoes even though fire on a kart is not a giant concern. So give fuel lines a once over every so often for security and condition.
Recurring problems are anything like a seat stay that keeps loosening. - Probably a bad fastener.
I like to check the front end (or other system) either from center out or left right/pairs. Once you get to where you can check a given section in a consistent manner it becomes an easy habit. Even if the steering feels fine it is very easy to give it a look and at least verify the safety clips are there and nothing is obviously bent. Look for a loose tie rod end, binding, bent parts, etc. Get a feel for how tight to have the axle nut. Are spacers even?
I like to pay attention to feel and sound. That way if a new feel or sound pops up you can maybe take it as a warning.
I would limit my session length at first both for your physical and mental stamina and to be able to check the kart at shorter intervals.
Clean your kart as you look it over is a good practice if you have time at the track. After it comes off the trailer and into the garage cleaning begins. I am the odd ball in that I will use soap and water on a kart. Some don’t at all. I like to get it clean quickly. You can try to keep the water off of the bearings and other parts that you don’t want to get wet. We tend to get rain enough here that it is going to get wet a few times a year anyway. I don’t go off track as much and in the past but that can be a factor.
Some of what I like to do in the garage:
Make sure lap sensor is pointing the right way for next track day.
Check frame for damage
Check seat mounting
Check Floor Pan
Check all nuts and bolts
Clean & lube rear axle bearings
Check Steering Upper Half
Are fuel lines hardening
Throttle Cable for chafing
Is carb aligned & fully opening
Brake rotor and gear for warping
These are not in order. Think about what the biggest issues are and address them first. So a cracked frame is a big problem you want to deal with ASAP as opposed to a loose bolt on a floor pan.
Checking things in the garage that are bigger and longer term issues are good while you have the luxury of time.
As some of example of what do before going out for a session:
Air pressure in tires
Always check brakes
Find an order for these - that way you can get ahead on them if you want. So you can add fuel early but you may want to adjust pressure right before going out. Doesn’t mean you can’t do it ahead of time, just mean you have to check/adjust it at the proper time again. Doing it ahead can spot a leaking tire.
I always check brakes late in the going out list as I race in rentals too or if I jump in someones kart I at least check the brake. In fact when getting into a kart putting you foot one the brake as you get in has it’s merit.
As far as front end geometry I would use the Tony kart baseline as a starting point. Then you can adjust from that. You can get the front end to where it handles weird and eats tires.
Spares is going to be something to build up.