Three places for boots - Waltons (in Belgano), Casa de las Botas (in Palermo) and I know someone here, Felipe Magnasco (Grand Prix rider) who also makes them. Waltons makes the more traditional styles, Casa de las Botas will allow you to pick and choose styles and Felipe does traditional as well as some very funky varieties (patent leather for example). Send me a message when you are here & I'll pass along the numbers.
As far as riding - you have 4 options in the city. You can ride at GEBA which is the smallest of the clubs and is public. They offer lessons but there is no membership fee. At Clubs Aleman, Buenos Aires or Hipico Argentino, you can take a month of lessons I believe but then have to join the club (200 pesos a month social membershp fee) on top of lesson costs. All are located in downtown (GEBA, Buenos Aires & Aleman) are on Dorrego btwn Alcorta and Lugones. Hipcio Argentina is on Alcorta, next to River Plate stadium. CHA (hipico) is the nicest but most expensive of the clubs, Aleman has the best footing and I haven't ridden at Buenos Aires (although footing for shows can be bad) so can't comment on being a member there.
Aleman and CHA also have pool, restaurant, gym, etc.
Let me know if I can be of any help.