"Omnivore" is a bit misleading
Most carnivores can eat some plants
eg lions sometimes eat grass (probably as a digestive aid)
Most herbivores can eat some animals
eg deer sometimes scavenge carrion
Most omnivores heavily specialize in either meat or plants
(really the specializations are a bit more detailed than that, with not much overlap between vertebrate and invertebrate consumption, and specialization among various plant parts such as leaves, fruit, seeds, tubers, gums and fungi)
Among mammals, the "true generalists" who do not heavily specialize in any one food source are almost all rodents. Virtually all mammal consumption of grain/seeds comes from specialized herbivores (eg ruminants such as cows and sheep) or from these rodent generalists.
Humans are definitely evolved to eat meat, especially large fatty animals. We have limited ability to convert protein into energy and so on a meat-based diet need ample fat. There are many nutrients in meat that we can't get (or can't get efficiently) from plant sources. Vitamin B12 is absolutely essential and cannot be had from plants (vegans have to supplement). Other vitamins like vitamin A are found in the human-usable form in meat but only in a precursor form in plants, which our body converts inefficiently (so "Vitamin A" is not necessarily Vitamin A).
Our shared ancestor with chimpanzees probably ate quite a bit of fruit. Modern fruit doesn't resemble wild fruit, and humans are evolved to suppress uricase and therefore struggle to break down uric acid which is a byproduct of our fructose metabolism (hypothesis: this aids in adding body fat during fruit season to survive the winter). So moderate fruit intake isn't the worst thing but will give you gout symptoms and tend to accumulate body fat, especially in the liver where sugars/alcohol are converted into fat. In my opinion the best fruits for human consumption are fatty stone fruit like olive and avocado. Semi-sweet stone fruit may work well such as peaches, cherries and plums. Berries may work well, especially strawberry which is high in vitamin C (to offset the negative VitC impact of sugar consumption).
Starchy roots & tubers (USO - underground storage organ) are handled OK, most will simply break down into glucose. Excess glucose for long periods will probably give insulin resistance, and there are certain deficiencies in USOs that some human populations are adapted to deal with and others are not. If you are descended from a population that ate lots of USOs you still need meat but can probably get a chunk of your calories from USOs without too much negative effect.
Certain human populations are adapted for dairy, for example continuing to produce lactase into adulthood (to break down lactose). You may be able to eat some dairy products (eg cheese, butter, yogurt) and not others (eg milk) or you may not handle dairy well. Keep in mind dairy is optimized for infant animal growth, so if you're trying to lose weight it may be a poor choice.
Finally we have little or no evolutionary history that would make insects, grains, seeds, nuts, leafy greens, plant bodies, fungi, gums or honey suitable foods.
