@LawrencePayne probably gave the most practical advice: don't give up when you hit the initial plateau.
To that I add: take advantage of your adult ability to think abstractly and self-evaluate.
TL;DR
I'm not a teacher so file this under: personal anecdotes...
I took guitar lessons as a kid. I never questioned what I was doing in those lessons. Zero music theory or improvisation was included. I learned how to strum and fingerpick chords, but I could have learned so much more.
Now, as an adult I'm studying piano on my own.
To me, the difference between studying as a kid and an adult is: as an adult I can think abstractly, self-evaluate, and research my own study materials and repertoire.
For example, I don't just learn the fingering for a specific piece of music. I try to understand the underlying mechanics and then look for its application in a variety of piano method books. I make up drills on such things in all keys, varying rhythm and harmony.
I do similar things from theory studies. When learning about harmony sequences, types of motion, etc. I will apply that in little drills in all keys.
The point is to identify musical elements abstractly and understand how they are used in different contexts. Relative relationships and abstract modeling are hugely important in music so there is a great benefit to studying this way. Kids don't think abstractly like that, but adults do. Use that to your advantage. You will get more out of your study time when you realize everything has some underlying pattern or model which gets re-used in countless ways.
Not too long a ago I experienced the pay off when trying to play Mozart's K. 545. Not the first movement that everyone learns how to play (I learned that about 15 years back) but sight-reading the andante. Not that long ago attempting a sight read would have been hopeless. But after a lot of pattern drilling for about two years I really was able to read the piece. At certain points my hands started playing things before I realized what I was actually doing! The basic mechanical and harmony patterns had already been trained into my hands. I mostly had to pay attention to the rhythm. It was very satisfying.
So, set your own goals, self-evaluate, and use your higher level thinking.
If you really are starting with zero musical background, take time to review and compare method books. There are tons of beginner methods at http://imslp.org. Don't forget about the famous Mozart and Bach notebooks.