Trainability has little to do with intelligence. Different breeds were developed to perform different functions and therefore handle the tradition methods of training differently. Beagles on the other hand, where bred to run in the field, independent of their owner, in pursuit of their prey. This in no way suggests that Beagles are less intelligent, they just work in a different way. All dogs will need their vaccinations, neutering, worming and exams.
Some breeds are more prone to health problems than others.
Investigate the breed you are considering and talk about any potential health problems with a reputable breeder. By going to an animal shelter you will be saving the life of your dog. Most shelters keep a list of potential homes and their requirements. Try to spend a little time with your potential new pet outside of the cage in a quiet area. It is difficult to observe even the best dog under the stressful conditions of a kennel.