While it's tough to protect an idea in an unprotected market, some suggestions are:
1) Positioning: Stake your claim and let everyone know that yours was the "world's first ______." Claiming first-to-market can impress some people in your target audience.
2) Alliances: Sure, someone can copy your idea, but if you're first and you form strategic AND EXCLUSIVE alliances with partners who can either add value to or help you promote your idea (or better still, do both), the copycats out there won't be able to deliver the same added value or do as as easily. It also helps if you can form a partnership with a market leader who already has strength that a start-up business doesn't have.
3) Promotions: Again, your competition can copycat the idea of promotions, but if your promotions are clever, unique, and "keep 'em coming back for more," you'll have a better shot at developing a loyal and engaged customer base. Those customers might like your promotions so much, they can even bring you new customers! Speaking of...
4) Leverage Existing Customers: Use your existing customers as a way to help spread the word (e.g. via social media and other referral opportunities), You can even consider some kind of incentive or loyalty reward for doing so. I have first-hand experience as a customer of Blue Apron, and they do this well. Every so often they "gift" me with a free one-week trial to give to a friend. There's really nothing in it for me other than feeling like I'm doing something nice for a friend...but that's not a bad incentive!
Hope this helps.