"I occasionally hear things about how SQL sucks and it's not a good language"
SQL is over thirty years old. Insights about "which features make something a 'good' language and which ones make it a 'bad' one" have evolved more rapidly than SQL itself.
Also, SQL is not a language that conforms to current standards of "what it takes to be relational", so, SQL just isn't a relational language to boot.
"but I never really hear much about alternatives to it."
I invite you to ponder the possibility that you are trying to hear only in the wrong places (that is, the commercial DBMS industry exclusively).
"So, are other good languages that serve the same purpose (database access) and what makes them better than SQL?"
Date&Darwen describe the features that a modern data manipulation language must conform to in their "Third Manifesto", the most recent version of which is laid down in their book "Databases, Types & the Relational Model".
"Are there any good databases that use this alternative language?"
If by "good", you mean something like "industrial-strength", then no. The closest thing available would probably be Dataphor.
Oh, and by the way, a software system that is used to manage databases is not "a database", but "a DataBase Management System", "DBMS" for short. Just like a photograph is not the same thing as a camera, and if you are discussing cameras, and you want to avoid confusion, then you should be using the proper word "cameras" instead of "photograph".
The Rel project offers an implementation for the Tutorial D language defined in "Databases, Types & The Relational Model", but the current prime goal of Rel is to be educational in nature.
My SIRA_PRISE project offers an implementation for "truly relational" data management, but I hesitate to also label it "an implementation of a language".
And of course, you might also look into some non-relational stuff, as some have proposed, but I personally dismiss non-relational data management as multiple decades of technological regression. Not worth considering, that is.