Lamb hearts lamb hearts lamb hearts. One of my favorite offal cuts. Used in stews you won't even notice the difference from more expensive cuts, but they are also great in couscous etc. A really approachable offal. Also, get the St. Johns cookbooks (Nose to Tail and Beyond Nose to Tail) and/or the River Cottage Meat book. Both have great offal recipes.

Chopped liver is some kinda heaven.

Saute an onion until it's soft and then add a bunch of chicken livers. When the liver is cooked through you can either literally chop it until it's a nice paste, or toss into a cuisinart.

Add salt, pepper, a bit of mayonnaise, and if you feel it, some chopped hard-boiled egg. Oh sweet country Jesus, is it ever good.

Try confiting chicken hearts or gizzards. Very little organ flavor to them. Chicken liver pates and mousses are also a good place to start.

I suggest you try recipes made primarily of meat and fat, with a tiny amount of organs. E.g. liver sausage composed mostly of meat and fat.

I agree about using heart and would also add tongue. They are very lean meats and aren't as extreme texturally (in my opinion) and are a bit more forgiving when cooking and still turning out delicious. The 'Nose to Tail' and 'Cottage River Meat Book' that lonelocust mentioned are great resources for offal.

I'd also suggest being offal adventurous when you go out to eat; offal cuts are appearing more and more in restaurants and you can start to get a better understanding of what these cuts can be like and things you might try out at home. I know it goes a bit against the economic argument, but even just one amazing experience out with some tasty sweetbreads/tripe/heart/liver/etc can really boost your motivation for your own cooking attempts.

when they have them, sweetbreads (@Cafe Juanita) and tripe (@Spinasse) are two of my favorites...

There is much to love about organ meats. It also doesn't have to be an overly difficult preparation. Key to beef liver is to not overcook. It cooks quickly. My family loves it simply. I heat up a heavy skillet to medium heat, add some olive oil (also healthy), toss in the liver slices (dashed with just salt & pepper) and let them brown. While I'm doing that, I'll saute some onions, then pour over the liver slices. My kids like to add potatoes and maybe some corn.

Oxtail is awesome in stew. You can also slow cook it until the meat falls off the bone. Use the meat as shredded beef in a multitude of dishes. Use the stock for gravies etc...

Tongue is fabulous smoked. Love it. Tongue is fatty. You can also peel the skin, slice it real thin & sear it in a heavy skillet & eat. Nothing fancy. The fat in the tongue is extremely rich. Another easy option for tongue is to slow cook it for several hours (8-10)in a crock pot. Then peel the outer skin. The meat will shred and can be used in fajitas, salad toppers, bbq sandwiches, etc.. or add to soup. Lots of options.

I'm not a fan of kidneys but we have several customers that love these.

Mountain oysters are delectable fried.

I strongly recommend the Japanese kushiyaki treatment for organs. My favorite stuff on skewers: heart, chicken gizzard, and beef tongue. And the best part is that it's cheaper than the conventional cuts.
Also, have you ever had pho with tripe? It's got a nice texture and blends in with the noodles.
I don't love liver either, but as a spread on bread or in banh mi it's pretty decent.

How are you with giblet gravy?