I have had many people ask me about whether they should apply for a job. Sometimes they say that someone else is more qualified or worse, that job is Suzie's job, she earned it and they don't want to take her job. Sometimes it's a feeling of devotion to their current boss.
Let me start with the later excuse. Your current boss, if he/she is a good one will understand and should encourage you to advance in your career. Trust me, they will survive just fine without you. Not that I am encouraging you to leave without deep thought, but that alone should not be the sole reason for staying if advancement is an opportunity. Moves are hard on everybody (even families) and you should weigh all aspects of a career change.
Now I'll address the thought of not applying because you feel it would be better for someone else or that someone else is more qualified. My mantra for my career has always been if you are qualified for the job, and you would enjoy the work, apply. The selection committee will determine who is most qualified. If you are ultimately not selected, you should be able to ask for feedback to see where you might improve for next time. Also, while you may not be the best qualified, the one the is better qualified may not apply or maybe even bomb the interview, leaving you to potentially work for someone that is less qualified than you are. That is definitely a bad situation.
I have been reading a little in John Maxwell's book which focuses on lifting your lids (the things that hold us back fro achieving greatness, see below for the link). Primarily he suggests, you have to lift your own lid first. Then you have to help others lift their lids, AND you must allow others to lift the lid you are not able to lift yourself. But common to us all -- we all have lids.
The 21 Most Powerful Minutes in a Leader's Day: Revitalize Your Spirit and Empower Your Leadership by John Maxwell
If you believe one of your lids is inexperience, lack of training, or lack of education, I say work on lifting those lids and allow others to help you as well (starts with a good mentor).
So apply on! Good luck. Ask for feedback and be a better employee first, then a qualified candidate second.
So you got the job. Should you continue to apply for more opportunities? Especially those that you feel qualified. My advice is to wait. My personal feelings is you should stay at least 2 years before actively looking. You don't want a reputation of not providing value to the organization that promoted you.