Try building a Zoo Cage. They come in all sizes. Now you can concentrate on catching the critters instead. Catch a critter with a Critter Cage, spawn a Zoo Cage, spawn the critter, hit it with a Zookeeper, and it stays forever as your pet!
To create the different sizes, start at the smallest and just keep adding to it to create one size bigger.
Materials: iron block, glass block, quartz block