I agree that it sounds like right now, improving other things before meat might be more important, unless you are ok with ordering online. In terms of omega 3's, sardines and anchovies are SUPER high in omega 3's. If you haven't already had them, they aren't as gross as you think. Well they can be to handle, but in the right amount they add a lovely depth to food (anchovies in particular).
In terms of shopping the grocery shopping, I strongly recomend getting your hands on a copy of the weston a price shopping guide. It will help you identify which brands use which meathods and thus what is healthier than others. (they're 1 dollar each including shipping and handling). I have an old copy, but for instance, without the shopping guide, I would never have imagined that organic valley butter is from grassfed cows without the guide. I would be totally lost when buying the few condiments I do buy without it. For instance San-J soy sauce and muir glen ketchup are available in the supermarket here, and are on the good list (it is best, good, avoid, both national (and a few local) brands, a lot of mail order brands, and also a description " good: pastureized naturally fermnted soy sauce; naturally sweetened ketchup; sauces with natural ingrediants and without MSG; pasturized vinegar" There are definitely a lot of small good brands that aren't on it (particularly some dairy and many cheese), but I find it helpful.
Definitely shopping the outside of the store helps. Mostly, I'd say its slowly changing your habits. Make soaked oatmeal, or coconut rice pudding or something weekly. (Cook some dried fruit in it, and serve with a fat (butter, cream, etc) and a natural sweetner such as molasses, maple syrup, honey, until you get used to it being less sweet.) Find something you would like your family to snack on, eat up the last of the cereal and don't buy more. Try not having any in the house and redirecting people to fruit, veggies, cheese, crispy nuts or some other snack. I'm trying keeping a bowl of blanched broccolli in the fridge for lunch and snacks and its great. Broccolli is cheap, it seems to keep for a good 3-4 days (longer than broccolli keeps in my fridge!), its tasty with some fat and salt, and I remember to eat lunch because I have something to eat) It would be even better if I had a container of aoili right next to it, but I haven't mastered making mayo and don't have the money to spring for a coconut oil one. (Mayo keeps longer if you add whey or yogurt cream I think, I might try making some and adding some yogurt).
Maybe if you have the time, money and gas, you could take a family day trip to the farmer's market (they're very fun) and buy a case or two of some stuff, and freeze or ferment it. A case of canned applesauce, a case of frozen peaches and a case of pickles? Otherwise, just do your best. Obviously, you can't do better than your best, and it will be good. In my opinion, striving to improve (in anything) is more important than trying to do it perfectly. So your only produce isn't super fresh and comes from far away. Cabbage will still make good sourkraut if it sat in a truck for 4 days, and it will still have lots of available vitamins and good enzymes and taste good.
Soaked beans are healthy, traditional and inexpensive (and thus a good way to help set off the fact that other parts of TF are more expensive). Try soaking them for 12-24 hours, (with a little baking soda. this helps keep the beans from seizing, or getting tough). Then make them sort of like pasta (in the sense of bring a lot of water to a boil with your beans but then let it simmer, you aren't waiting for all the water to absorb.) Taste them and you should be able to tell by the texture when they are done. Then serve them in various ways with fat (lentil salad, refried beans, a pat of butter on top....). It also depends on what kind of beans you use. I like french green lentils for lentil salad because they hold they're shape and don't turn to mush (many lentils do turn to mush). I also like navy beans for the same reason, and I like their mild taste.
This is a ton of overload though, and you'll probably do better going slowly. buy real butter instead of the country crock (grassfed if you can get it but if you can't, any real butter with no additives, organic or hormone free is good, but anything is an improvement), and any whole milk that isn't ultra-pasturized. horizon organics is, as are anything labeled UHT, or with a sell by date way in the future. (these milks are shelf stable but no good for you). Once this is normal for you, then you can try something else. Change your cereal habit, search out better animal foods.
Little things like trying recipes that uses fatty fish or sardines or anchovies can be done whenever it doesn't feel intimidating. (Canned salmon is perfectly acceptable, and good as salmon cakes.)
good luck. hth and doesn't overwhelm you. congrats on deciding to improve your families health and eating.
Traditional foods are fantastic. They taste so good and make you feel so good. In terms of shopping at the grocery store, it's totally workable: stick to the perimeter of the store and shop heavily from the the produce aisle, for egg's make sure to get omega-3 eggs if pasture-raised aren't available. Don't be afraid to buy foods online, either. I think grass-fed meat is really important and unless you can find that locally, online is your second best bet. Instead of cereal, try making a huge batch of baked oats and work from there. Soon you won't even miss the cereal.
I found this link to be really helpful when I was feeling overwhelmed.
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