Is there a way to tell what biome I am in in Pokémon Go?

I think you may have a slight misconception about how exactly biomes work - biomes aren't a property of a specific large geographic region, exactly. Rather, Pokemon Go's map is full of invisible points called 'spawn points' that spawn Pokemon in an area around them based on a timer, and each spawn point has its own 'biome' property assigned to it (roughly based on real world properties of that location, though you get a lot of variation with how accurate that is) that determines what Pokemon it can potentially spawn.

People will talk about living in a specific biome if they are somewhere where the majority of the local spawn points are of a specific biome type - so, for example, if you life on a lakeside with water spawn points all around, you might say you live in a 'water biome'. However! That doesn't mean you can't find spawn points with other biomes nearby - for example, the local bus stop might still be an industrial biome, or a nearby park might have a common biome, and so on. From what you say about seeing Eevee, Abra, Bellsprout, and Squirtle nearby, it sounds like you have a multitude of spawn points with different biomes around you.

So, what does this mean? Well, basically, that finding the best biome for Dratini means paying attention to where exactly the different kinds of Pokemon spawn around you. Dratini's normally seen as a rare spawn from spawn points assigned a water biome type, so the fact that you've seen Squirtle around you is a good sign. What you'll want to do in order to have the best chance of finding dratini is make note of where you see water Pokemon spawning most often - most likely near real word bodies of water - and check back in at those locations often.

It may still not be especially quick to get a dragonite, since dratini is a rare spawn (and, from reports, has become rarer since Gen III, with more and more potential water biome spawns giving it competition for spawning), but if you check in potential areas regularly you should find dratini much more quickly than simply roaming randomly.

Unfortunately there's no overall trick to determine biome, since it's determined per-spawn point, but if you search '[your location] Pokemon Go spawn points' or similar, it's often possible to find local guides written by Pokemon Go players from the area detailing the best places to go for finding spawn points for each biome type.