Welcome to the forum!
The first step is really careful tracking of spending for at least 3-6 months. I just do it in excel, recording down every single expense on sheets I've created for each month, but others do it using fancier software or apps, like YNAB. There are some spreadsheets stickied at the top of one of the forums here (Called "One sheet to rule them all" or something like that.) It takes awhile to get into the habit of recording expenses, but it is well worth it.
I agree with HipGnosis that a emergency fund of a few thousand should be built up as a priority. However, I also think you should sign up for your employers 403b/401k plans as soon as possible. If you are not used to having much money, then I say pick a pretty high target for your contributions (10-15%? more?) Make sure that you can adequately cover your expenses with what is left over. If you can't, then you can always reduce the amount going towards retirement next month. I did this with my first job, and older me now is very thankful to younger me!
A great book for just starting out with saving and investing is called Millionaire Teacher by Andrew Hallam. It's easy to read, probably because it's written by a teacher, and your local library might have a copy for you. ARebelSpy recommended it to me, so I am passing on the recommendation. Good luck!