I'll preface this by saying I'm no gymnast, and we've only just started using them in the gym in the last few months - but my eldest was a gymnast and has offered us some great advice and coaching. I am not giving this as coaching advice, just as a lifter sharing experiences. It's not something I'd have a novice barbell lifter mess with, but if an intermediate wants to play a year or so in, why not, they've earned it.
I'll also say I'm not sure it adds even a pound to the actual barbell lifts. But they feel better - your current lifts feel better and more stable. This could just be a novelty effect. We'll see.
That said, when coming from barbells, the ring stuff is a matter of technical progression. As in, "first do X movement, then you can do Y movement." With barbells we're used to it being simply a matter of finding the right starting weight and load progression for novices. The technical progression isn't really there for slow lifts, only for quick lifts. But for rings, it's there from the start.
For example, you'll see the upright support performed, where basically the person is upright with their hands by their sides. At the start the person mightn't be strong enough for this, particularly if they don't have a good barbell press and/or they're heavy. So you do a prone support - basically you set the rings up at a height of knees to ankles, get into a pushup/plank position but with your hands on the rings, and hold with your arms extended. For some particularly deconditioned people, or those with a history of lower back pain, they might need to start in a knee pushup position. Once you can hold it for 30" you can progress the technical difficulty, from knees, to toes, to an incline support, to an upright support. In each case, arms in line with your body (ie roughly a barbell press grip width) and squeeze the legs together.
It's similar with other movements. I'll outline the progressions we're trying, again noting that we're just experimenting, and this isn't what gymnasts do because they start as kids and are only doing it for gymnastics, rather than for variety and to complement barbell lifts or their lives in general.
The movements are broken down into push, pull, trunk (aka core), and legs. The legs part is the most experimental because it requires the most co-ordination, which is more unusual in my gym population than brute strength. Hey, we're a barbell crew.
The technical progression here is that one is the prerequisite for the rest. The static holds we want to see for good sets of 30" before attempting the next progression. The ones with reps, 12 reps. So for example you need to be able to do 12 dips with your feet supported by the bench before you go to the bands, and do 12 banded ones before you go to the ones without supports.
We do a testing session to see where the person is, and see how many seconds or reps they can do. After that we halve those numbers and go for 3 sets, and build that up at 3" or 1 rep a time. For example one guy easily did the upright support for 30", so we looked at his dips and the bench was so laughably easy we didn't bother with testing banded, went to full dips and he could do 2. His work sets were then 1,1,1, next session 2,1,1, then 2,2,1, after that 2,2,2 and so on. Some things like pushups can be progressed faster. With the pistol squats the issue has been getting a below-parallel squat, most of us find it really difficult.
PUSH
Supports
- rings at ankle height, on toes, prone support
- rings at knee height, on toes, incline support
- rings at shoulder height, upright support
Pushups
- rings at knee height, on knees, pushup
- rings at knee height, on toes, pushup
- rings at knee height, feet elevated on bench, pushup
Dips
- rings at chest height, feet supported by bench behind, dips
- rings at chest height, knees supported by bands, dips
- rings at chest height, no support, dips
PULL
Rows
- rings at hip height, knees bent, supine rows
- rings at hip height, knees straight, supine rows
- rings at hip height, knees straight, feet elevated on bench
Chinups
- rings above head, negative chinups
- rings above head, banded chinups
- rings above head, chinups
TRUNK (aka "core")
Rollouts
- rings at knee height, toe pushup position, feet on rings - bring legs towards body then extend back again
- rings at knee height, knee pushup, hands on rings - begin with arms pointing down, straighten them out to as close to parallel with the ground as you can, bring them back
- rings at knee height, toes pushup position, hand on rings
Leg raises
Prerequisite: upright supports
- rings at chest height, upright support position, bent leg raises
- rings at chest height, upright support position, straight leg raises
- rings above head, hanging support, straight leg raises
LEGS
Squat, assisted
- rings at shoulder height, split squat rear foot off ground
- rings at shoulder height, pistol squat, off leg going forwards
Hinge
- rings at hips, using both hands, one legged hinge, ie with knee slightly bent, pivot forward on one hip reaching the other leg back without raising the same hip
- rings at hips, using one hand, one legged hinge
- rings at hips, lying supine, one legged bridge (aka hip thrust)