You can use your own tiles. I tried 2 things. Generating my own Mapzen vector tiles (they use same format as Mapbox) And you can also use them for free. Here is their layer descriptions. This is quite work intensive. You need to have postgresql and load whole OSM PBF export into the database, then you run python server which requests data from this database and renders vector tiles. I think it is meant to render all the tiles in queue since it took couple of seconds per page to render visible tiles. Most of the time was spend in python after DB server was queried.
It's advantage is that you get nice tiles back. It has basically everything you need, but is much harder to customize. For example if you want to add specific style to cycle ways. You need to go deep into the code and change couple of query templates and a lot of other things.
Then I tried Tilemaker. This is just C++ program which reads OSM PBF dumps and lua config script (where you specify what tags to send into tile) and spits out mapbox tiles. It's advantage is that it is much easier to set up and customize and that all tiles are rendered at once. But it is harder to create nice tiles. (AKA load all the different highway tags are roads just of different kind. It is up to you to specify that but this already works in previously mentioned Mapzen and also Mapbox).
For example kind in Mapzen roads layer In mapzen this is already taken care of but in tilemaker it is up to you to write all the conditionals that get road type from different OSM tags into a layer. And it gets more complicated in landuse tags since kind is a:
combination of the landuse, leisure, natural, highway, aeroway, and amenity OSM tags, or urban area and park or protected land for Natural Earth areas.
Of course you can have completely different tags but it is nice to have one which tells you what landuse you are looking at.
You ned to know that Mapbox, mapzen your custom mapbox tiles all use same format, but each will have different tags. So the style you create for one probably won't work for the other.
For creating styles you can use Mapbox Studio (but is probably useless since it is in public beta currently and I'm not sure if you can specify own tiles there).
I used Mapbox codeflow, which is basically nodejs server with gulp script that reloads site with a map when style file changes. It also supports writing styles in toml, JSON5 and yml in addition to JSON. It also shows errors kinda nicely. (only line numbers are missing) Currently it support version 7 of styles but 8 is currently out. For getting line numbers of errors I used Mapbox GL style spec which can also update style to the new version.
You can also try Glug which is a different style language which compiles to Mapbox GL style. It is a little more compact.
For using tiles you can also create mbtiles with mb-util and use them.
