When watching professional ukulele players on YouTube your attention doubtlessly is focused on the musicians' hands and fingers as you're trying to see how the music is made. You probably won't even notice the subtlety going into the act of holding the instrument, another important aspect of instrumental play requiring special skills and practice.
Before learning to play the uke I've read about the unique circumstances of holding a uke, and the part the instrument's size plays in its successful mastery. Of course, playing a uke with a shoulder strap can eliminate this problem altogether. Due to its small size, the uke doesn't present players with an obvious way to handle it during play, unlike the guitar (or the piano :). This became clear to me very quickly the first time I held a uke in my hands. You have to make sure the uke doesn't crash to the ground even while your left hand is busy supporting the neck and fretting, and your right hand is busy with strumming. My usual mantra works for this scenario as well: It's all just a matter of practice. Assuming short daily practice sessions, a beginner player can get comfortable with holding the uke within a few weeks. Give it a solid month and you'll barely waste any energy on fidgeting with the positioning anymore. From personal experience I can tell you that playing while standing is extremely difficult, except of course if you have a shoulder strap. Sitting on a chair, or perhaps sitting Indian-style, and propping your uke in your lap is the easiest way to hold your uke. Sitting allows you to rest the uke on your upper thigh, so you can relax your grip at all the contact points. (When I was a beginner I even tried to rest the bottom of the uke against the arm of a chair.) So, as the best option, find a comfortable chair and sit. Don't get too comfortable though, leaning back and putting your feet up just isn't going to work. Wedge the uke between your right elbow, forearm, and upper thigh in such a manner that your forearm only touches the soundboard, but not the strings. Angle the uke away from your body just enough that you don't muffle the sound and destroy the acoustics. Left handed players will have to mirror everything described above. Once again, I strongly discourage beginners from playing while standing up without a shoulder strap.
Looking around on YouTube among all the ukulele videos, you will find that the musicians employ many different kinds of strumming techniques. Traditional ukulele strumming is mostly done by the index finger, but some fancier strumming techniques use the thumb and the other fingers as well. Picks (especially the commonly available guitar picks) are not popular with ukulele players because they click the strings and sharpen the sound, none of which enhances the uke's lovely, unique sound. On rare occasions when a pick is called for a felt pick, made specifically for ukes is used.
The simplest strumming, done by the thumb, is mostly used during tuning and while learning/practicing new chords. To play songs and to vary strumming speed, use the basic strumming technique using the index as shown in the video below:
Point at your chest with your right index finger, rotate your wrist downward, and let your index finger strum each string as your hand travels down. Now practice, practice, practice! The perfect combination of the right angle and right speed will sound the strings in unison. There are a lot of requirements to fulfill during strumming, and usual, practice makes perfect! Strumming the right way is not terribly hard, with a little practice you can get it right within a few days. The exciting part comes when you start repeating the strumming motion rhythmically, thus giving the base character to the songs you'll play.
Soon we'll start getting to know the the chords!