Here are the bare bones of moves you should try to achieve: a handstand, cartwheel (one armed if you can), a scale and a front roll. This is something anyone can learn unless they have severe motion sickness in which case you will need to omit any inverted motions.
The best idea is to go to an adult drop-in gymnastics class and learn about the moves and what they are called. A coach will demo them and you can decide if you wish to try this or if it is even feasible for you. The coach can work with you on this decision.
Give yourself at least a year to learn any new gymnastics before you begin to look at competing, otherwise work with what you CAN do and make them clean and polished.
Leaps
Practice some of your leaps on a trampoline. It is great practice and many gymnasiums have tramps which you can warm up on with your straddle and pike leaps! This would be fully coached. Trampolines can be dangerous.
If you dabble or are experienced in gymnastics then I suggest you take this score sheet to your coach or an adult class and get graded. With this information you can choose your stronger moves to add to your routine. If you want to use one which needs polishing then you can focus that into your training.
Some of the skill names can vary among coaches. Many of the holds are more known in fitness, but your coaches should be able to grade you on your attempt and you should be pretty familiar with most of these.
Holds
Progressing yourself to be able to achieve some basic holds is not too difficult and can be done at home! Once you get good at them you can add resistance such as ankle weights.
Pike Holds ///
Pike holds are a common fitness routine element and can be varied in the end result to show some creativity, but first you have to be able to do one! You should incorporate V sits, hanging leg lifts in full range of motion and some pike leaps to help you toward this hold.
Beginners can start seated with your legs out in front of you, toes pointed and thighs tight. With your hands at the sides of your thighs, nearer the knee for a balanced lift (but as you get stronger you get closer to the hips so you can lift heels), lift your tush off the ground by arching your back, pulling up on your pelvis and if you're able then get onto your fingers - lift your palms.
Heels lift on the ground but if you are not at that level you work toward that. Some people are unable to bend their fingers in the manner to get palms up and get some height clearance so for those I suggest using fists for better height than using flat hands. Hold this for 5-20 seconds repeating for 2-5 sets total.
Straddle V ///
Advanced can go into a straddle V from the pike hold then back into the pike hold. Toggling your feet is also a cute effect while in a pike hold. Lastly you can bend a leg while in the hold then alternate bending your legs, meanwhile holding yourself up, which shows impressive strength. Heels together or apart, up into a side V is also nice.
From sport aerobics you can try the pike rotation turning a full 360 degrees on hands only. Men's gymnastics also has a move where you go from a pike hold directly into a handstand but this is extremely advanced!
Straddle Holds ///
Straddle holds are another common routine element. Like the pike hold, once you learn this you can get creative with it and create new holds. Like for the pike holds, by incorporating V sits, hanging leg lifts in full range of motion and some pike leaps, you make it all easier.
You try it with your feet off the ground, using a bench or chair. This allows room for heels to sag without touching, until you get stronger.
At this point you can begin to practice on the floor and should be able to get your hold even if only for a second. Practicing these with sets and reps as above will get you to a strong hold on the floor. You can try this with one hand behind you, this leads to a straddle turn.
Try it with your hands behind you so you can lift heels! Advanced can go into a straddle V from there back into the hold. Lastly you can bend a leg while in the hold then alternate bending your legs, meanwhile holding yourself up.
Next step is one done in gymnastics which also has a move where you go from a straddle hold directly into a handstand but this is extremely advanced! Coming into a straddle from a handstand is much easier.
Yoga Holds ///
Yoga Holds have a variety but this is the basic one. Once you learn this you can extend a leg or vary your position to be more creative.
Crouch down with hands to one side, flat on the floor, keeping the inner arm elbow into your hip fold, while the other elbow supports your outer thigh. Slowly lean into this and try to balance, taking your feet off. As you bobble during your learning, you can use your toe as support to prop you back up.
This is an easy hold to get so just keep doing the above, trying to hold it longer each time. Practice this for 5 mins 3-5 times per week alternating sides so that you can try to get both sides. Try playing with variations!
Split Holds ///
Split Holds are easy if you are flexible. You get yourself into a push-up position; put a leg up onto your shoulder then lean forward to let your back leg up. You can bend that leg too if want to add flair. This is another hold you have to just do, trial and error! It's all about balance.
Planche Holds ///
Planche Holds have a variety of creative poses. You should practice holding your legs up while balancing on your hands with your feet apart. Using a body ball to get you going is a great start, then soon take that away and try on the ground, using elbows to prop you by putting them into your mid section.
As you get stronger you can do some with one of the elbows at your side. Once you can get up and balance then take your chin off the ground. Next step is to try the leg variations. The lower horizontal planches are hard and used in men's gymnastics, some men spin on one arm!
Dance
Taking a few dance classes is a great way to keep up with hot new moves if you are not in the clubs regularly or watching videos all the time! Even if you have previous dance experience but have not been into it lately, this is a great advantage toward sharpening your stance, poise, form and movement.
A must for anyone with no dance experience! Hip-hop, latin dance, jazz and break dance are great dances to begin with and commonly available at reasonable drop-in fees. You want to keep it all high energy and these are high-energy forms of dance!
You want to refine any movement you decide to put into your routine, polish it and own it! Some common transition movements are chasses, pirouette, step slide, grapevine movements, step turns and more!
High Kicks ///
High kicks are basic dynamic moves which always spice up a fitness routine. They can be easily achieved as cool moves to fill in spots where you wish to show your incredible muscular endurance and flexibility. You need to make sure you can bring your leg up without dropping your butt or bending the knee in the anchor leg.
Chest must be out and shoulders back and relaxed. It impresses the judges if you can do both legs. Hamstrings and groin should be sufficiently warmed up and stretched.
Push Ups
Once you can do a push-up, try doing them with one foot off the ground and knee bent to make it look pretty! Another idea is to do a push-up, touch elbow and knee together then push-up and do it again on another side.
From here you can try moving your push-up sideways so you do a push-up then cross arms and legs to move toward the left and repeat. When you cross legs, bring the leg under the other, not over.
Yoga Twist Pushups ///
Yoga twist push-up takes the twist hold into a push-up, first practice getting the hold then you can try bending the elbows to get a push-up or 2!
Hinge Close Grip Pushups ///
Hinge close-grip push-ups are done by keeping elbows stuck to the body in the push-up, which is actually a triceps pushup! So you bend to 90 degrees then slide back onto the elbows, return and up.
Hinge with legs out is done like a regular push-up but feet together and at the down phase your elbows bend toward the body to touch the ground - while that happens your feet split out with a hop, then back in when the elbows return.
One-Arm Hinge ///
One-arm hinge is when you hinge your elbow in during 1-arm push-ups down phase, but I think this one looks sloppy so I usually don't suggest it - it is no stronger than a 1-arm push up, which is seamless and smooth versus adding the hinge.
One-Arm Pushups ///
One-arm pushups start with a negative rep. Try slowly lowering your body to the floor using one arm beginning at the up phase of the push-up. Try to place your anchor hand in a balanced position so as to avoid pressure on the shoulder joint.
Again, as in the regular push-up, make sure you keep your belly button pulled into your lower back (this is called being 'engaged') and do not put your butt in the air or drop the belly! Keep feet shoulder width apart and hands out beyond your shoulder area.
Do not drop your chin, try looking up a bit. Elbows should bend to 90 degrees, chest leading. Some girls start with their feet far apart to learn and get stronger, which is fine but you must slowly work them closer so that by routine time your strength shows and you don't look like you're cheating with your feet too far apart.
Split Pushups ///
Split push-ups are easy if you're flexible enough. You simply put a leg on a shoulder and do your push-ups. Nice transition is to go into the split hold. Some strong girls can do this with the anchor leg off the ground hovering!
Straddle Pushups ///
Straddle pushups are also an awkward looking push-up which I do not favor but it seems the spectators and judges do. This is done with both legs on your shoulders as you do the push-ups.
One-arm One-Leg Pushups ///
One-arm, one-leg pushups are fabulous and come from sport aerobics. This is all about balance and taking your regular push-up up a notch by bringing the limbs inline then slowly lowering. Add in the up phase as you get stronger. A towel around the waist is a great tool for a partner to help pick up the slack with.