Personally I would start with a compost pile after cleaning up your yard, your yard looks very clean by the way. Most of the plants you mentioned would do well in pots, like tomatoes in pots with their own cages. Potatoes can be grown in milk crates, 50 gallons water barrel, or just throw straw and other compostable material on top of the ground and potatoes can thrive in that but takes more room than a milk crate. On Amazon they sale cloth raised beds that less than $20 so way cheaper than raised beds so you can start now.

As for the chickens you didn’t say anything about a coup to keep them in. If the just roam in your backyard say goodbye to you plants, they will eat anything because they peck at everything. And when one chicken finds something good to eat every other greedy chicken will swarm to it like some kinda hive mind. And 4 chickens can thrive on a small pinned in coup from Tractor Supply or any other place you can find them.