Level 2 is usually automatically reached after 5 days. Level 3 after 22 days and so on.
Animals can be leveled up faster using Vita Gems, levelling a critter to level 2 generally requires 2 vita gems. Levelling one to level 3 generally requires 9 vita gems. Levelling one to level 4 generally requires 75 vita gems
Other than paying 75 vita gems to level 4 from level 3, there is a second way. The second way is to wait until December the 26th. Tip: be sure to ready some level 3 cirtters before Christmas!