If you shop at Hispanic and Asian markets, the prices are much lower than at regular supermarkets (Ralphs, Vons, etc.).
Hispanic markets have very good prices on all types of dry beans, and rice. Vegetable and fruit selection is good.
Asian markets usually have an incredible selection of vegetables and fruits, at very good prices.
When I got my first apartment in the mid-1990s, my grocery bill was only $10 to $15 per week. I was already a vegan, which helped. Also, I lived in an African American community, where leafy green vegetables were very inexpensive (3 big bunches of mustard greens for $1).
Here are some inexpensive dishes that I used to make:
Boiled/nuked potatoes with a little salt or ketchup
Lentils and brown rice
Soup: Lentils, onion, mustard greens or kale, canned tomato sauce (not spaghetti sauce, which is too sweet for soup), with soy sauce and red pepper sauce to taste
Stir fry: Onion, lentils or tofu, celery, and mustard greens or kale. Homemade sauce made from soy sauce, white vinegar, and ginger (many recipes available on internet)
Spaghetti with lentils and canned spaghetti sauce (sometimes cheaper than jar spaghetti sauce)
Natural peanut butter on brown bread (this is more expensive)
Here's what I learned from that time:
1. Dry grains are way less expensive than pre-made grain foods (like bread and cereal)
2. Large dry beans take a looong time to cook. That's why I stay with lentils - they require no pre-soaking, and they cook in 45 minutes or less.
3. Homemade sauces are easy to make, and are way less expensive than pre-made sauces
4. There's nothing cheaper than potatoes. If you shop around, you can find 10-pound bags of russet potatoes for $1 or $2. Just remember that potatoes are low in calories, so remember to eat plenty!
Potatoes are dirt-cheap. Here's an interesting video from Washington State Potato Commissioner Chris Voight, who ate nothing but potatoes (with a little oil and some condiments) for 60 days. He did this in order to prove that potatoes are healthy, and surprisingly nutritionally complete. In this video, he buys a week's worth of potatoes for $7.50: