I'm thinking about making the switch to a mechanical keyboard, but I don't know much about them. Anyone have good recommendations on a starter brand that's good quality but won't break the bank?

Corsair strafe is fairly "cheap" and still packs the delicious cherry mx switches :)

Another good one is the Logitech g610 ones :)

*note these are from personal research and preference.

My first mechanical keyboard was from Qisan, real cheap and sturdy, works great and I still have it :) definitely recommend it. Check for the kind of keyboard spring that you think best fits your typing style and you is set ;)

Just keep in mind that while great to game, cherry mx red switches are hard to type accurately with due to their lack of feedback. My current favorite are razer's green switches, which are similar to mx blue if I'm not mistaken.
Also, mechkeys are really loud, so you might want to install o-rings if noise becomes an issue

I would recommend something with mx blue switches.

May be a bit pricier to some but I'd recommend the SteelSeries apex m500 with blue switches, I have a red switch version and love it, miss my blues