Try using double 00 flour and mixing it with semolina.
Watch
Pasta Grannies on YouTube, there is a whole collection of recipes and ideas. When I get bored, I make strozzapreti.
If you use just semolina and water, you're not going to see much difference between that and store-bought, simply because store-bought is semolina and flour. Every time I've used AP and eggs, it always tastes fresher.
You do need to realize they have different rules about preservatives in food, even flour, in Italy, and that affects how things taste. People always remark everything tastes fresher in Italy, part of that is because it is fresher. It's the difference between cooking food you bought from the market earlier that day and cooking food hauled to you by Sysco or US Foods.
That said, every time I make pasta using eggs and AP flour, I can immediately taste the difference.
The first time you make pasta, it can seem like a lot of work, but it really becomes easier the more you do it. Making the dough is easy, just toss the flour and eggs into the mixer with the dough hook. Wrap the dough ball in saran wrap (or a bag) and put in the fridge (or set aside on the counter) for 30 minutes to allow the flour to fully absorb the eggs. Take dough out of bag, cut into sections, send through the pasta machine, starting at the widest setting and working to second-thinnest, putting it through twice each to develop strength in the noodle. Then either cut it with the machine, or use a knife, your choice.
Make your sauce from scratch, too. With you're starting out with fresh pasta, I'd recommend a lighter sauce, like aglio e olio or cacio e pepe, so you can taste both the sauce and the pasta. When you're happy with how your pasta tastes, I'd move on to a basic pomodoro sauce.
They don't tell you this, but the pasta attachment is perfect for anything that requires rolling width where you'd normally use a rolling pin. It saves on labor and makes for an even roll. I use a manual pasta machine it to make cannoli shells.
