"Almak" has a broader meaning. It also means "take/get". If someone gives you something, or borrows you or lends you something, then you are taking it.
Kitabı arkadaşımdan aldım - here it is more like borrowing
Parayı annemden aldım - here it is more like getting money
Adresini şirketinden aldım - here it is more like learning/getting. I got your address from your company.
"Satın almak" is buying something with money from a shop etc. If, in the context, it is easily understandable that you paid for getting that thing, then you can omit the word "satın".
Kendime yeni bir ayakkabı (satın) aldım(I bought a new pair of shoes for myself. Since shoes are new, we know you wouldnt get it from somewhere else but store)
Marketten süt ve yumurta aldım (I bought milk and eggs from the shop. We know that you gave money for that.)
Sevgilime bir hediye (satın) aldım
Bu saati internetten (satın) aldım
If there is a possibility that you borrowed it, then please use 'satın' to avoid confusion