One of the first and very important strength skills a gymnast learns at the intermediate level is the straddle press handstand. You start from a medium-width straddle stand, put your hands down just in front of your feet, and (without "jumping") lift your legs through a wide straddle up to handstand. Sounds simple, but it requires a good deal of abdominal and arm strength as well as good flexibility in your straddle.
To begin, work on straddle-up to handstand. Begin in the same medium-width straddle position with your hands about a foot or so in front of your feet. From here, go ahead and bend your knees slightly and push off your feet to pass through straddle and up to handstand. This is considered jumping, but is fine for now while you get used to the movement. The key here is to push off your feet so that you can get your hips over your head, getting your center of gravity right over your hands. Once you get to this point, make your straddle as wide as possible so that you continue to keep your center of gravity over your hands.
When that becomes easy, try doing a straddle press headstand. Start in the same position you were in before, but go ahead and put your head down on the floor as you would for a regular headstand. Now you don't get to jump off your feet -- you have to use your muscles to slowly lift your feet from the floor up to a straight headstand. Also try to straddle back down to your starting position. This should teach you how to lift your legs slowly.
Then, try doing a straddle press headstand with your head up on the end of a panel mat. Your hands should be on the floor just in front of the panel mat, and your feet starting on the floor, just a little bit further from the mat. Most likely, you will have your back rounded so that your feet can touch the floor. Now as you lift your legs up to headstand, you will have to straighten out your back -- rolling your hips, as we say -- so that your hips go over your head. This is a small but very essential movement to learn before you can do a press handstand.
Now, try doing a straddle press handstand with your back against the wall. Put your hands on the floor about six inches or so in front of the wall (depending on how tall you are), straddle your legs just a little wider than your shoulders, and then lean over with your shoulders past your hands so that your upper back rests against the wall. Then, lift up on to your toes and try to push as much of your back against the wall as possible, then flatten your back to get your hips against the wall. By now your feet should have lifted off the floor and you should be able to lift all the way up to handstand. For most people, it is this first part where you are rolling your hips over your head that they have the most trouble with. You should feel a lot of pressure against the wall because you are leaning so far into it, but when you do a press handstand for real without the wall, you would open out your shoulders and push out to a straight handstand.
Another essential technique you need to learn how to master is the initial lift of your feet off the floor. This requires a good deal of hip flexor strength and, again, abdominal strength. Start in the press handstand position again, lean your shoulders over your hands a bit, and just slowly lift your feet a couple inches off the floor. Return to your starting position. If you can, try to hold yourself in that position with your feet off the ground for a few seconds. Think about pushing your hands into the floor and pushing through your shoulders.
Another way to work on developing those abdominal, hip flexor, and, to a small extent, your arm muscles is to do handstand straddle-downs. The first couple times trying these, have someone spot you by standing in front of you and holding you at the hips. Kick up to a straight handstand, then as slowly as you can, straddle your legs and lower them. Push up through your shoulders and try to miss your feet as you lower yourself into a straddle-L hold. As you do this, you have to lean your shoulders over your hands to keep your center of gravity over your hands. You also should pull your legs toward you, being in more of a straddle-pike position as your lower into the straddle-L.
Now you can begin trying the straddle press handstand with your feet raised on a panel mat or other raised surface. Having your feet elevated makes the first part easy for you -- your hips should be already over your head or very close to it. You still need to lean your shoulders slightly over your hands and make sure you hips stay over your head as you lift your legs up to handstand. Go up onto your toes as high as possible, onto your toenails if necessary. Again, you should begin with your feet just a bit wider than shoulder-width, but as you lift your legs up to handstand, try to make your straddle as wide as possible. Here is where people with a flexible straddle can take advantage of it.
If you are using a panel mat, unfold it one panel at a time as this becomes easier. Eventually you will be able to do these from the floor, but as you get lower, you have to put in more effort into rolling your hips over your head before lifting your legs.
Once you have mastered these and can do ten in a row with no problem, you can try a more advanced version of this skill -- starting from a straddle-L hold. One way to learn the rolling motion of starting from a straddle-L hold is to do it along the length of a panel mat. Start lying on your back on the panel mat, lift your legs up to a candlestick position, then roll forward and straddle your legs and put your hands in between them on the panel mat. Here, push up as high as you can into a straddle and keep your legs lifted as much as possible (keeping them close to your body). You should be continuing to roll now, and use the rolling momentum to try to press up to handstand. At first, you may not be able to press very high. In fact, your feet may not lift up past the panel mat. But keep working on this drill until you can roll all the way up to handstand
When you can do that, go ahead and try doing the press handstand from the straddle-L. The key here is lift your legs up as high as possible as you start to press, so that you have room to clear the floor while you rotate and open through your shoulders. As you can imagine, this requires a lot of abdominal strength.
One popular place that press handstands are used are on beam. You can either do them from a straddle-L crosswise with the beam, a straddle-L sideways with the beam, or a straddle stand sideways with the beam with your hands on the beam. It is this last one that most people begin with. The tricky part here is that your hands are starting in line with your feet, making it more of a challenge to balance on your hands and getting your hips over your head. You may need to make your straddle more narrow to get your hips closer to that position. Also, you may need to go back to doing a jumping straddle handstand until you get accustomed to doing these on the beam, then you can try pressing up. Once you can do that, you can work on trying to do it from a straddle-L.
There are many other ways that you can do press handstands: from a pike stand, from a planche, on the parallel bars, on the rings, etc. And they are a good precursor to many skills, such as straddle cast handstands on bars. They are also certain to be an impressive addition to your gymnastics repertoire. Good luck!