Indy is 100% doable on your budget - I know because I do it more or less! I LOVE Indy and will sing its praises.
We have a few great local trainers, but we also have clinics year round (Sharon White, Tim Bourke, the Laws, to name a few), trainers from KY who come up and teach once a month or more, HTs at the Hoosier Horse Park, plus a schooling series on the northside. If you want more competitions, people go to KY, OH, MI all the time. I ride year round with an indoor, minus the few truly awful days. As a city, I love it too. All the good amenities, great airport (is it weird to be obsessed with your city’s airport?), traffic isn’t bad at all, and if I, a longtime southerner, can live here weather-wise, truly anyone can. We have an awesome local group of eventers and it’s a huge reason Indy has become home for me. Also @SeeSpotEvent has the most beautiful badass spotty ponies you get to drool over.
Definitely more on the northside, although I think there are still some options on west and south side. For reference, I live in a far north suburb (20 miles almost exactly north of downtown) and it takes me 35 minutes to get to work downtown during rush hour, when we do that whole ‘go to work’ thing. I’m 15-ish from my barn. Most people my age range (mid 20s) don’t live in the 'burbs like I do, but I prioritized house value, barn proximity and safe neighborhood and this is where I ended up and it doesn’t bother me at all. If you want to live right in Broad Ripple (fun, lots of bars, walkable), budget would be stretched, ditto downtown.
For reference, I’ve lived in Alabama (…um), Atlanta (traffic is miserable, summers suck, expensive), Greenville, SC (getting pricier, traffic on 85 made me want to cry and die simultaneously, worse airport, I found board and lessons to be WAY out of my price range), RDU ($$, traffic, general dislike of the city) and Albuquerque, New Mexico (uh, don’t move there if you ever want to event again).