Some of the best brands of sneakers or trainers that you can get include Adidas Originals, Lanvin, which were founded by couturier Jeanne Lanvin in 1867 and New Balance. Others include Laventino, Raf Simons, Balenciaga and Saint Laurent. You can get a list of the sneakers from http://www.fashionbeans.com/2013/10-key-mens-trainer-brands/.
Adidas makes good trainers, my personal favorites for running shoes, ultralight. Feels like feathers on your feet. Nike are good too, but every pair that I've had the soles come off. New Balance are pretty good also, wear these a lot, have a couple pair that are several years old, and still in good shape.
If you are thinking of running it is very important to find the right model that fits your feet. Wrong shoe causes wrong stepping which causes damage unfortunately for many people. Nike, Adidas, Puma, Asics and Reebok have no difference. General purpose shoes with mid-quality. In some models Adidas is wider and Reebok has more comfortable walking shoes. They are cheaper and less durable but it is good for you to change as much shoes at the beginning preferably from different models so you can find the right one. If you want better shoes you should look for New Balance, Brooks or Saucony. NB has more models than the others. Brooks is the most durable brand. Saucony has the lightest running shoes. But they are a little bit more pricy.