Hi there and
to the forums!
Congratulations on your little one! First of all, although it seems like he's settling in well so far, usually it's recommended to let your budgie settle in for two weeks before trying to take him or her out of the cage. This is so that they can build a sense of security in their cage and get used to their new environment without the added stress of coming out! During this time is a great time to get to know him by sitting next to his cage and talking to him as often as possible, etc. so that when the time comes to work with him on taming and bonding he's more used to you
From what you've described, I don't think he's sick since he's active and not showing any other symptoms. Budgies normally do quirky things like that, such as closing one eye, and the beak movements you're seeing are probably him adjusting his crop after eating or for a stretch, or yawning! Of course, if you're concerned, it never hurts to discuss things with your avian vet to make sure that your little one is doing ok.
Meanwhile, you've come to the best possible place to learn even more about the best of budgie care practices! It's great to have you with us - to start, be sure to read through the forums' many budgie articles and resources, including the many Budgie Articles and "stickies" (threads "stuck" to the top of each subforum for easy reference) to ensure you're up to date on everything.
If you have any questions after doing so, please be sure to ask as we'd be happy to help!
We hope to see you around!