Marvel's Avengers microtransactions will give you several ways to pick up new cosmetic items, though you'll be able to earn plenty just by playing through the base game.
In a news post on the game's official site, Square Enix breaks down how Marvel's Avengers approaches gear and cosmetics. When it comes to picking up new gear, you'll be well acquainted if you're familiar with Destiny: you can equip the stuff you find while doing missions, or you can break it down for more resources to spend at vendors that you meet throughout the course of the story.
The cosmetic items come in four types: outfits, emotes, takedowns, and nameplates. Cosmetic vendors will carry weekly and daily rotating selections of goods, and you can pick them up in exchange for the standard Units currency you earn while playing.
The Marketplace is where you go to spend real money. It's accessed from the main menu of the game, and everything there is sold for Credits (which come in a minimum pack of 500 for $5). The Marketplace will carry a weekly rotation of cosmetic items that you can pick up a la carte.
Then there are Hero Challenge Cards, which you can think of like battle passes for each individual hero. Each card has 40 tiers of free and premium rewards, and you can earn Challenge Points toward unlocking them by completing daily and weekly challenges.
The six heroes available at launch will have their premium Hero Challenge Card tracks unlocked for free. You'll need to spend 1,000 Credits to unlock the premium tiers for post-release heroes such as Hawkeye, and you can earn all 1,000 Credits back as rewards from the card. Then you can put that toward your next premium unlock, and so on.
Square Enix has also confirmed that Marvel's Avengers will have exclusive cosmetics on PS4, and free skins for PS Plus members.