I definitely recommend a program, because consistency is key, and a program will give you something to do each day, with some nice variety between days to keep things interesting. DAREBEE has several core-focused programs for which each day's workout is quite short. So they can easily be added on to whatever else you're already doing.
The beginner friendly core programs are:
Back & Core
Total Abs
2-Minute Abs
Of the above, I think Back & Core would be the best choice to help you achieve your goals, because it has the greatest balance between abdominal work and back work, and also has yoga days and days with standing work. All of these things are key to helping one achieve better posture. (Total Abs also contains some standing work, and both Total Abs and 2-Minute Abs include some back-focused work every fourth day. But Back & Core has the best balance in terms of focusing on all the key areas needed for good posture.)
In terms of individual workouts, I like:
and:
You'll note that the last two in particular are upper-back focused, not abdominal focused. In our modern culture of desk jobs and hunching over cell phone screens, a weak and overstretched upper back is a common culprit in poor posture (often more so than weak abs and lower back). If you were to choose one of the more strictly abdominal focused programs, I would definitely recommend pairing them with some extra upper back work like these.
Back & Core really does give you everything though. If you want a one-stop shop for increased core strength and improved posture, Back & Core is my top pick.