My order of big purchases in the early game typically go like this:
Day 3: Fiberglass Rod and some Bait
Day 4: Large Backpack
Day 7: Upgrade Pickaxe to Copper
Day 11: Upgrade Pickaxe to Steel
Day 13: 20-40 Strawberry Seeds, plant 20 on the Speed-Gro from the Spring Crops Bundle
Day 18: A Coop* with 1 Brown Chicken, 3 White Chickens
Day 20: Harvest the first Strawberries, upgrade Axe to Copper
Day 24: Upgrade Axe to Steel
Final Days: Big Barn, Deluxe Backpack, the first few Vault Bundles, depending on what I'm aiming for. Making sure I save at least 10,000G for Summer Crops (100 Hops with Melons and Blueberries in between, plus a few others)
*Like
@Lew Zealand said, the Coop is great because the Mayonnaise Machine recipe is unlocked at Farming Level 2, whereas the Cheese Press recipe "needed" for Cows comes at Farming Level 6.
But I started playing Stardew more than one and half years ago, so I've developed a route for the early game that I find works for me. These are in no way requirements, just how I like to play the game!
Hope this helps, have fun!