I'm tired of wasting time grinding smithing, so I'm doing a new character. I bought them steel, but its starting to get to the point where I'm going to need something stronger. What are some good dungeons that I could go loot delving for better armor? Or am I better off just saving my money and buying it when I can? Also should I bother with the alteration armor spells since I'm not using smithing?

Dwarvern ruins have a good amount of loot, you have a good high level chest at the end usually.

Bandit Chiefs usually have a full or partial set, though only in the quality of Iron, Banded Iron, Steel and Steel Plate. With Dragonborn, they also have Nordic Carved. The Reavers(Solstheim version of Bandits) often have Bonemold and Heavy Chitin too.

Alfthand has a Steel Plate Cuirass, Boots, and Gauntlets at the end of the Dwemer Dungeon that simply requires killing the previous Owner.

Markarth has an interior Dwemer Ruin with a small Quest. The Dwemer Ruin has a full set of Dwemer Armor behind locked doors. Talk to Calcelmo, the Court Wizard to begin the Quest.

The Cursed Tribe Quest for Malacath rewards a Player with possibly a full set of Orcish Armor, Volendrung, and allows a Player to freely enter Orc Strongholds. The price is killing a few Giants.

Killing Bandit Bosses is a good source of Heavy Armor, but the equipment is mixed and tends to be based on character level.