Suggest you post this over on the Companion Planting forum here. It isn't something practiced by very many experienced gardeners who tried it long ago and then moved on. So as a new gardener, please take much of what you read about it with a grain or two of salt, ok? But the folks on that forum love to talk about it to new gardeners.

Companion Planting forum

Dave

Thank you for the feedback Dave!

companionship is part of soil management together with rotation and cover crops and attempts to get more food out of the same area. In fact you will have to try things for yourself.

consider for example an association of tomatoes and cabbage, as is suggested in many sites. I tried in Michigan and it does not work, or at least produces subpar cabbage while probably giving me also less tomatoes. But it is possible that in sunny Arizona that will work as the amount of sunlight there is enough for both, while tomatoes protect cabbage from excessive sunlight.

You can reason your way into several successful companionships by following some rules. First, you want to optimize sunlight collection. You can plant lettuce with tomatoes, they will be gone by the time the tomatoes grow to get big. Even when big, some vegetables produce dappled shade. Sunlight that hits the soil is wasted and is in fact a slight negative. Some greens make it in such conditions. In various beds, I have chicory, arugula, corn salad, and clover (to grow nitrogen), under tomatoes and asparagus. Note that very often these small, shade tolerant veggies are mentioned in companionships due to their ability to withstand partial shade. You can evaluate canopies and take your own decisions. I have not found any companion for collards, zucchini, cardoon and potato. They just cover the ground so completely you can not grow anything under.

Second, you ought to minimize root interactions. Deep rooted and shallow rooted veggies use different resources, effectively allowing a more efficient use of the garden. Carrots go down to 8 feet, which is double the tomato depth. They will be using different nutrients. You can find a description of roots of veggies here

http://soilandhealth.org/wp-content/uploads/01aglibrary/010137veg.roots/010137toc.html
http://soilandhealth.org/wp-content/uploads/01aglibrary/010139fieldcroproots/010139toc.html

and you should have an idea of the underground canopy in your garden. There is really no gain in ignoring it. For this reason many suggest carrots paired with tomatoes, but I am dissatisfied with carrots ability to cover ground. After the spring greens are gone, I have either clover (medium-rooted) or chicory (deep rooted) under toms. Chicory gives me three crops (spring, fall, and spring again after winter), and effectively I get more food from the same patch. and chicory completely closes canopy.

Third, there are principles of soil health that you can find in hundreds of webpages or youtube presentations from the USDA NRCS, ATTRA, SARE, Rodale Institute and others. Mixing different crops, and in particular having mycorrhizal species, has long term benefits for the soil. I mix carrots and kale, and parsnips and lacinato, plants that are similar in aerial size, with different roots and different microbiology, and it works fine, even though I lose the next spring kale crop when I dig the carrots. Beets and beans, most roots and onions follow the same above ground/below ground/mycorrhizal considerations. You always have to figure out that you will pick the roots after the other crop of course.

Having living mycorrhizal roots continuously also helps the soil (overwintering roots), and deep roots are crucial for soil infiltration (water moving down after rain) and capillarity (water moving up when it is dry). All this is accomplished not necessarily with companions, but also with winter cover crops or rotations which are part of the same soil management concepts. Be aware that some cover crops are tough to kill. You could also just make at least part of your garden a cottage garden, with many species mixed and permanent roots from herbs, and see if it works for you. in a cottage garden it is tough to grow root crops without hurting nearby roots, so I don’t do it.

List of plant associations with myc.:

http://mycorrhizae.com/wp-content/uploads/2013/03/Mycorrhizal-Status-of-Plant-Families-and-Genera-PDF.pdf