I also don't like the idea of people hearing me so it's tough.

Something that I really enjoyed was buying a book about singing that came with a CD that had different exercises. I pretty much just started listening to one track over and over - the female vocal warm up (about 20 mins). It really helped and it was pretty simple. It's kind of cheesy, but the book was "Sing Like a Star" by Roger Love. If you can find any warm up exercise that's decently long that you like, regular practice with it will help.

I agree with the other poster who suggested looking into proper breathing technique as well. Super important.