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You can rest assured that all of us made those awful cat-scratch sounds at first. I have a couple of suggestions. For what you are aspiring don't get hung up on playing in more than 3 or 4 keys at first. I suggest sticking with D, G, and A. Those are the easiest on fiddle and also the most likely to be used if you wind up playing along with a guitar. Maybe add the key of E also.
If you are taking lessons your teacher will probably have you practicing broken thirds which I think are really good. I also like the Suzuki book's exercises to improve intonation. I think it refers to that as "tonalization". Your teacher will surely know about that.
These have helped me a lot.
@tinksquared I think you will do just fine if you practice and focus on what you would like to play, as mentioned above. I agree it helps immensely to memorize songs. I recommend jigs and reels, as they are short and sweet, quite repetitive, and forgiving. They roll from one to the next. May not have lyrics, but they can be a lot of fun and quite simple. As a fellow-newbie, I found after practising a bunch of them over 6 months or so, I sat down one day and tried practising them without the book. It works great! I had to look back here and there, but now I know a handful of songs off by heart.