You can do the following:
-Let it hold a Soothe Bell. It boosts the happiness gained from berries and massages.
-Feed it certain berries (Grepa, Hondew, Kelpsy, Pomeg, Qualot and Tamato).
-Level it up.
-Feed it Vitamins.
-Massage it in Veilstone City.
-Walk 128 steps (50% chance of raising happiness, only works if Budew has 0-99 happiness)
-Battle against a Gym Leader, Elite Four member, or champion.
-Teach it a TM or HM. (only works if Budew has 0-99 happiness)
-Use a battle item. (only works if Budew has 0-99 happiness)
Make sure you don't do these things, because they'll lower Budew's happiness instead:
-Let it faint.
-Let it survive Poison with 1 HP outside of battle.
-Use Energy Roots, Heal Powder, Energy Powder, or Revival Herbs.
How to check happiness:
-Visit a certain NPC in Hearthome City.
-Use the Friendship Checker on your Poketch.