I don't know how much of this you already know. You can read up on the subject. Many people have written amazing works on this. I will just summarize a little bit of my experience with you in the hopes that it will provide to be a shortcut to avoiding many future frustrations that I had and still have to go through (because I did not follow my own intuition and knowledge from early on and engraved bad learning habits and habit is hard to beat out of your system). I realize you might be too new for all this but here are some things to keep in the back of your mind, they will become useful to you pretty soon when you start working on harder pieces. It's all very easy in the beginning but it's important to learn how to study pieces. It will save you time that you can use to enjoy your music more and work on other things.
As you've found out yourself, time spent in front of the piano does not equal good lasting progress. Mindlessly repeating a piece or parts of it will not get it in a presentable shape, at least not quick enough. What you will find later on as you get to more complex pieces is that your current approach does not work. Before you learn to smoothly play the piece you will have lost months on it and you will be discouraged. By repeating mistakes you learn them and then it's hard to learn to play a piece properly. That's why you need concentration and a targeted approach. Make every repetition count, don't just repeat 10 times for the sake of it. Know what went wrong each time and each time work on improving it.
So work out things that work best for you and follow some sort of routine. What I found works best to keep it organized is to have a little journal. Separate the piece you are studying in sections that make sense to you and write down which bars/measures those sections belong to. Analyze the piece in the simplest manner, no theory knowledge needed. See which bars repeat, see if you have anything in the dynamics and other markings that is unknown to you and look it up.
Start memorizing the piece right from the start. Work on small sections (2 bars is good, 1 if it's hard). Learn Left Hand then learn Right hand and be able to play it by heart each hand. This will take 5 minutes, no more for each little section. Then put them slowly together. Then go to the next section. When you are done with it go back and connect the two sections and work that way till the end.
Each new day you will have forgotten a little bit here and there. That's ok, go as often as you need back to the sheet music. After a few days like these with proper revision those sections will be concrete in your mind. You will have no more trouble ever with them. Build up the piece that way.
Always try to memorize as much as you can, not just the notes but also the dynamic markings. It's easier than if you try to put them on later because you'd have to unlearn the manner in which you played so far and add expression. Play musically from the beginning. It's not just "I'll play the notes now and when I can play them then I will make music". No savor every sound and listen. By listening to the quality of tone you produce you will gradually train your mind into controlling your hands so that you more often than not produce the desired sound. Over the years it will become second nature and you will no longer be playing just notes, there will be depth and people will like listening to you and will want to hear more.
Memorize fingering. If you are a beginner look for editions of your piece that includes fingering. Follow it best you can but if something is terribly uncomfortable make adjustments. Write down fingerings for sections where it can be confusing and every time you play follow the exact same fingering. If you don't you will confuse your memory because you will play in a slightly different way each time, making it harder to properly memorize the piece.
Even after you are playing it smoothly hands together don't rush with the tempo. Play at a comfortable tempo. But from time to time challenge yourself and play as fast as you think you can (up to the performance speed not over). Just when you do that don't repeat a section too long with that tempo unless you are playing with no mistakes each time. If mistakes start to pop up, relax, it's normal. It's good that you tested with higher tempo. This will help weed out trouble spots and is good. Do it often so you can always know where things go wrong and fix them.
Slow down those sections and work your way by gradually increasing the tempo.
After you get a piece up to speed, invest some time each week or every few days if possible and take your time and go through the piece slowly and listen, really listen. Refer to the sheet music to make sure everything is properly played and you are not hitting wrong notes or skipping something or playing with fingering that is making it hard to play smoothly.
If you don't revisit pieces in such a manner it's very easy to start making mistakes in spots that you played flawlessly. This is because when you play fast your brain doesn't have time to think much, you are on autopilot and over time your knowledge of the piece can deteriorate slightly.
There is more but I think that's enough information. I wish my first teacher knew how to explain these things so that I wouldn't have to find out the hard way. I would have made tremendous progress by now.
Just as an example I will tell you what happened to me over the past 6 months.
I have been assigned a dozen pieces probably and I spent hours upon hours every day. I woke up at 8, went to the conservatory, spent till 2 PM there in classes. Then I go back home and I play with lots of 10 min breaks till 9 PM.
That's every single day (minus weekends when I only get 2 hours each day).
Let me tell you that despite all my effort I never mastered any of the pieces because my practice was not organized.
I have good scores on my exams because my teacher counted in all the hard work and dedication and not so much the degree of mastery of each piece. But I know it was my fault and I know I did a ton of work but did it the wrong way.
So I was discouraged and I stopped playing for about 20 days. I just picked up again two days ago because I have to return to classes in 20 days from now.
So I have been practicing 10 times less. However, now I follow this little system I briefly described. I've played a total of 6 hours the last 3 days.
I picked up my last piece which I never got past page 1 during my study. It's Debussy - Doctor Gradus ad Parnassum. Amazing piece. It can be played extremely fast or at moderate tempo and is still adequate. I am aiming for something in between. Try as I might I spent a month and a bit over it before and I failed to play it smoothly in any tempo. The last 3 days I've been working it in sections and been concentrated and started memorizing right away. The piece is 50% memorized by day 3. So imagine what you can do in 1 month of such good practice. This piece will be done and over with by then and I can enjoy playing it. In comparison this piece would have taken me 3 months just to get going smoothly by practicing 7 hours a day the wrong way.
Secondly, when first starting out, until you learn a piece in a method book is it ok to just read the finger numbers, assuming a five note song which I will only have those for a while, until you learn it and then read the notes as you play or will this hurt you in the long run? What about those note charts too? OK to use them for the first couple of weeks? I am speaking of the ones that fit behind the keys and name the note and show it on the staff.
Best to learn all the notes and their positions. It's easier than you think and takes about a week to learn them and be able to quickly identify. A month of dedicated work on it and you will never again wonder "what's this note and which key on the piano you press to produce it".
I made flashcards for myself when I started. They had the note on a staff and the name of it in the back. I'd shuffle them and try to guess. I became able to identify them really fast in no time. Also try to imagine where it will be on the piano. It's not hard, just takes time to sink in, you can't cram that much info in your head at once. Take it a step at a time.
I also learned an online application on a website to learn the positions of all the notes on the keyboard, took a couple of days and a few weeks to translate it to a real piano.
Scales... Scales will be extremely useful to you in developing your technique but only if someone shows you how to play them properly. It's too long to explain here, once you get a teacher and when the time has come for that ask about it. Scales will be of no use if you don't know what you are doing. I can get into the theoretical side of it and their importance in building up your ability to play everything else, however it will be of no use to you as it does not directly translate to practical knowledge. You need to down to business and do it before that but before you start make sure you are doing it right. A good teacher will be able to help you and show you how scales are used. How your hands should behave when playing scales and what to look for in playing scales. If you are already playing scales then why not try putting the hands together by playing the scale of C in contrary motion? (Start on C with the first finger of either hand, not both though, then use the usual fingering, left hand will go down to the left and right will go to the right and the fingering in both hands will be completely the same, they will mirror each other, much easier than the other ways of playing them. This will get you acquainted in playing with both hands).
Just so you know, C is harder than some of the other scales it might not be the best starting one. E is better and actually much easier. D is also good.