So I got this brilliant game idea for a horror/suspense thriller and unfortunately, BYOND doesn't meet the standards to actually bring my plan to life. I was wondering if there are any engines that would be a good stepping stone where I could develop a first-person game with decent-quality graphics? Free or not, I would love for some recommendations! Thank you!

You seem to have trouble just making a BYOND game - "branching out" to something totally new when you can't even finish a project here is a great way to waste time. Anything outside of BYOND is going to be more difficult. I've tried, programmers a lot more talented than me have tried and we all come crawling back to BYOND where everything is sweet and simple.

But since I know you won't take any advice and insist on learning the hard way, try Unity3D. You can also make 3D games in Game Maker but I'm sure you have heard of that already.
I've created several games in my career on BYOND, however each being outdated and replaced by a newer.

Granted only one of the three projects wasn't a fan game. /:

Regardless, thanks a lot, bro.
The issue wouldn't so much be the programming (although it's fair to suggest there are added complexities) so much as the art assets.
listed engines that makes development easier:

Unity - Good engine with amazing graphics, but alot of practise to master

HLS - editing program for a game called "Amnesia - the dark descent"
Play it, and you will be amazed. with a few downloads and researtch, you can make preatty good custom storries!(Tho it is not a engine that creates groundup games, but it is fun and works!)