i try to buy Building Health 1 almost immediately to stave off early tower losses. can help the early game EXP race alot if you can manage to keep your border towers up for a bit longer with this upgrade.
the next one i go to buy is usually Building Damage 1. two reasons for this. first is that it grants AoE damage to tower attacks so enemy DG's won't be able to screen themselves from tower hits by standing next to minions or creeps. second, its cheap as hell.
i'll sometimes try to get Creep Health 1. this can slow down enemy farming of your creeps and also will make your creep waves beat enemy creep waves with a little health left over so they can naturally advance the battle line up the lane a little bit. every little bit of territory helps.
I personally like to get Experience Upgrade 1 but its not as popular as it deserves to be. often money is tight in close games so you might be forced to choose between this and Building Damage + Creep Health (similar cost, about 1100 for that vs 1200 for Exp).
Obviously Gold Income 1 is extremely important and you should get it at war rank 3 if possible.
after that you don't need many upgrades at all. most will just save up gold and wait for war rank 8 to buy the big creep push (priests, angels, and catapults all at the same time).
Once you've bought the creep push its pretty important to upgrade their health and damage so they charge to the citadel fast enough to prevent your enemy from stabilizing and farming them out.