Hi Natalie, and welcome to the forum!
It won't take you long, as you search around on this forum, to see that the vast majority of us find mozzarella to be the hardest of all cheeses to make. We can't figure out why it is billed as a beginner cheese - unless maybe because it is billed as giving an edible result in 30 minutes (by some recipes). I should hasten to say that there are a few folks on this forum who seem to have solved the puzzle and can get consistent results. I'm not one of them.
Some factors at work here:
1) In general, cheese is sensitive to the quality of the milk - and "quailty" here primarily means whether and how hot it was pasteurized, and whether it was homogenized. Both processes do enormous damage to the proteins and fat in the milk, making it much more difficult to get good results. Note that ultrapasteurized milk (which is often the case with organic brands) is so damaged that you pretty much can't even get a curd at all - just mush. What you are describing with a clean break, but then the curd "shattering," sounds like you are using regular (not ultra-) pasteurized and homogenized (P&H) milk. While you CAN make decent cheeses with this milk (all but one of my 110+ makes have used P&H milk), you will NOT get the idealized experience that is described in books and recipes. Curds will tend to shatter, "flocc tests" (which you may read about) will give very unclear results, and so on.
2) Again, you can still make decent cheese even with shattered curds. But the challenge of mozzarella goes way beyond the quality of the curds. The key is that mozzarella only stretches at a very narrow, specific range of acidity (around pH=5.3). When you do an acid-added recipe (e.g., the "30-minute" varieties), you have to add just the right amount of acid - which will depend on the characteristics of the milk. The traditional method of making mozzarella uses a bacterial culture to "ripen" the milk/curds; this process generates acid over time, so you have to catch it just when it hits that narrow window. In either case, you will benefit from having a good pH meter. However, you can experiment with trying more or less acid (or more or less time) to try to get it zeroed in. Someone recently shared success with the acid-added method ... after trying 12 times to get it dialled in correctly!
So, what's the alternative to mozzarella for a beginner? If you want to use acid rather than cultures, your best bet is ricotta or something like paneer. If you would like to take the plunge into cultures ... there's a wealth of possibilities, and we'd be happy to offer our favorite suggestions!