The Short Answer
If you take payments in person (retail, mobile service), start with Square.
If you take payments mostly online (invoices, booking flow), start with Stripe.
Why Square?
Square wins on hardware. Their card readers are cheap, reliable, and look professional. If you are a mobile detailer, a food truck, or a market stall, Square is the default choice.
Why Stripe?
Stripe wins on flexibility. It integrates with almost every booking tool (Calendly, Cal.com) and website builder better than Square does. If you send digital invoices or want clients to pay when they book online, Stripe is usually the smoother path.
What about fees?
They are basically the same. Roughly 2.9% + 30¢ per transaction. Don't waste time trying to save 0.1% when you are starting. Pick the one that fits your workflow.