How to complement a dish?

You could certainly get a cookbook of side dishes and learn some new favourites, but what might be more useful is a book that teaches you about flavour combinations and menu planning. I would suggest Culinary Artistry as one such book. It's not about specific recipes (you can find those elsewhere or make them up), but rather it addresses the kind of skill you're looking to learn.

One technique is to consider the origin of the main dish and choose side dishes from the same region. Seasonality or market-based shopping is another approach to expanding your repertoire.

Make meat "not the main feature". Having it as the main feature is a piece of history when meat was the most expensive part of the meal, and it also seems to have become an unhealthy piece of history.

Meat is significantly cheaper nowadays, so you can spend more on other things and make them the main part of the meal, not just side dishes.

It is a mental paradigm shift.

Start looking for new and unusual ingredients from small or local farmers, not just the staples from the main grocery stores, there is an entire new world to discover.

Try growing or wild harvesting your own ingredients.

Then the fun begins, finding recipes or techniques to use these new supplies and make that masterpiece.

The meal becomes much more interesting when there is more of "you" in it.

I'm a big believer in contrasts in meal planning.

For example, one thing I like to consider is the basic flavor profile of the main dish and the side dishes. For example, if you've got a particularly rich tasting main course, you could complement it with a slightly bitter side dish such as Brussels sprouts or broccoli raab. A sweeter vegetable would be a nice contrast to a more sour main dish.

Textural differences are also valuable. A stew or soup (which is generally all mushy stuff) works great with a crunchy side dish such as a nice crusty bread.

Think also about how the plate will work. If you've got a main dish with a gravy, you don't want anything on that plate that will not work with the gravy. Thus if you also want salad, give a side plate or serve it as a separate course.

The visuals of the plate are also a consideration. If you've got chicken in a cream sauce, mashed potatoes, and cauliflower, they're all close enough in color to make for an unappetizing meal. Color differences make things more appetizing -- that's why bright green parsley and bright yellow lemons are often used as garnishes. That's also why some dishes are visually appealing right away. (Arroz con Pollo generally has yellow rice, red pimientos, and green peas. Stir frys often try to include something red or yellow to contrast with the green of many of the veggies.)

Having said all this, what I'd recommend is taking the side dishes you like and start sorting them into categories. For example, starches can be divided into mushy, chewy and full of texture, and crispy. (Potatoes can be mushy or crispy depending on the preparation.) Then as you're planning your main course, think about how to contrast it in a pleasant way.

There are tons of grains, which are great for sides. (Bulgur, Rice, Barley, Millet, etc.)

Your additions to these grains could be simple, or complicated, according to your taste and time. Of course, these aren't the only things you can use to start a good side-dish. You could use pastas such as couscous or orzo. You could use polenta, as well.

If you don't want to add a vegetable to the sides mentioned above, consider adding a vegetable somewhere in the meal. You could do this with a salad or a simple steamed vegetable with butter.

Cooked rice, pastas, cous-cous, salads... a lot of variety to choose from. Even simple bread will do it.