so this is my opinion. I am not a cave instructor, but I am an active/experienced cave diver.
I did a 0-hero course in 8 days so there was no diving between. I was trained properly before hand, so there were no fundamentals that had to be taught and also had reel skills which helps.
I'm excluding any and all course work and lectures for reference, and assuming you have passed something like a GUE Fundies type class where you are at least a halfway decent diver. Unfortunately for most, Cavern/Intro is basically intro to tec/fundies with a bit of cave diving thrown in. I hate this, don't agree with it, so I'm going to pretend like you have your sh!t together before you start and is how I would run a cave class and is how some instructors run them if you are a good diver. This is also pointed at Florida cave diving where there is really no cavern zone to speak of in most of our caves, so doing a formal "Cavern" course is basically a waste.
videos of all of these skills here. I don't know Bil, some of the skills aren't executed as well as I would have liked, but the skills are all there and show you what to do. They are not a substitude for proper training obviously, but if you're curious, they are here.
Bil Phillips
Cavern/Intro
Skills: running of primary line, single file air sharing, blind air sharing, blind exit.
Day 1: Skills overview. If this is your first time with the instructor, which it likely is, they will have you do reel skills and then 0-viz skills in open water to make sure you can follow a line properly, preferably with touch contact. On land before you start you will go over things like primary/secondary tie off's and how to do them, and different line wrapping skills for proper line placement. Light communication, touch communication, s-drills, single file air sharing, and if in sidemount valve feathering can be gone over. This is a BIG day and will set the tone of the rest of the week for you. Depending on your location this may take place only in open water, or a combo of cavern/open water, but can all be done in OW.
Day 2: Basically another skills day and should be the start of cave diving where you go into the cavern zone and basically are practicing running your primary line into the gold line. Expect to lose your gas and all of your lights to make an air sharing, lights out exit. This is annoying and you will be frustrated, but you'll get over it. Depending on the cave profile, what size tanks you have, how good your SAC rate is, and how far back the gold line is, you may not get much past the tie in point because running line sucks a lot of gas for most new divers. You'll do this as many times as you have time for in the day.
Day 3: same thing, running the primary line to the gold line, tying in, and going into the cave until you hit your gas limits *this will depend on the agency and instructor*. Assume that at some point during the dive all of your equipment is going to fail and you will be coming out blind while sharing air. Expect a lost buddy drill once on the gold line, as well as lost line drills.
Day 4: repeat and complete
Apprentice/Full
Assuming with same instructor so they know your skills
Skills: learning how to "jump" off of the main line, learning about complex navigation, and decompression in the cave
Day 5: jumps: learn how to tie in a jump reel, cookie/arrow placement and how to navigate through them by touch. You should know this because you already learned how to tie into the gold line for your lost buddy drill where you can see, and you know how to tie the reel onto the gold line from your lost line drill as well as tying the primary into gold line. So combine the beginning of a lost buddy drill, with the end of the lost line drill, and voi la, jump. Expect to lose gas and lights and exit blind.
Day 6-8: You will basically be doing the same thing but with more complex dive planning. Circuits, traverses, etc. No new skills really, but you learn how to properly plan a circuit and traverse which require a bare minimum of 2 dives to do properly but typically 3 *you only get 2 if you are lucky on where the lines end up for clean up. One in, place cookie at thirds, one "out" and if you hit that cookie before you reach turn pressure you can complete the circuit/traverse, and then a third to "clean up" depending on how the cave is set up. Expect to have lights out and air sharing here, and include decompression planning and execution.
For the NAUI style 2 step plan, day 4 is replaced with day 5, and day 5 is included in the first course. GUE I believe is similar, but have slightly different limits. I much prefer this style of training since a 5 day course which allows you to make 1 navigational decision while diving to thirds of your gas is actually enough for the majority of cave divers, while Intro limits of 6ths with no navigational decisions is far too restrictive forcing divers to go to full cave. This is my opinion though so take it for what it is worth.
To your specific questions.
Gaps=short jumps. Technically supposed to be done the same way, but some are actually so close that you can use a double ender bolt snap to connect the two. Rare, but they exist. Example here is the jump into Kings Bypass in Jackson Blue.
Circuits are a complex dive plan where you "jump" off of the main passage into a side passage, and that side passage connects back to the gold line at another location. A super "complex" dive plan is a very common dive in training at Peacock Springs that combines a traverse and a circuit called the "crossover". Technically you are making a traverse because you end up in another sink-hole, but since you can't enter at that sink hole, it has to be treated as one dive. Anyway, the plan would be to enter into Peacock 1 and go up the Peanut line, place a jump into the crossover tunnel, then jump out onto the gold line in the Olsen tunnel. You can choose to go into the Olsen tunnel until you hit thirds and place a cookie, turn around and come back, or you can pop out at the Olsen Sink and then turn around. Dive 2 is go into Olsen and if you arrive at your jump reel before you hit thirds, then you can complete the circuit. Some instructors will allow you to "clean up" on your way out since you have a verified line back to the surface, others won't. I prefer to clean up on the way from that dive.