I have for you here a lovely picture of Tiny Tim, one of the greatest ukulele-powered entertainers of all time, showing quite clearly how to properly hold a ukulele. Please note that Tim is holding left-handed in this picture - so you can line it up as though you're looking at yourself in a mirror. (Tiny could, and would, play either way!)
Note that the uke is cradled by the right arm - it is also squished between your wrist/forearm and body. The index finger of the right hand is extended, and is used to strum the strings, right at the place where the neck of the ukulele meets the body. Guitar players want to strum near the hole - don't. Part of the nature of the ukulele's sound come from the strings being strummed way up there - right in the middle of the part that will be vibrating.
The fingernail of the index finger faces the ground, so that the back of the nail strums the strings on the way down, then the tip of the finger and nail strum on the way back up.
The ukulele is held high on your body - anyone who held a guitar up so high would look foolish. This position allows you to partly hold the uke up in the crook of your elbow. It also helps you to hear the ukulele while you are singing, when it is closer to your head. Your right forearm should point right up the uke's neck.
If you are holding the uke properly, you should be able to take either hand away and have the uke stay where it is. That's easy enough for the left hand - just grab the neck between the heel of your hand and your fingers, as though you're making a chord. It's a little trickier for the right hand - you have to keep the whole hand free for strumming, so you have to get used to sort of cradling/squishing it in there.
Give it a try. It's worth getting used to hanging on with just the right arm (not the hand!), otherwise when you are playing songs, the uke will keep catching you off guard by slipping, and breaking your concentration.