Allergies would certainly be the most common cause of generalized "itchiness" in dogs. It can cause all of the things you've described above, including itchiness, redness, rash, bumps. I'm going to start off by giving you a brief description of the different causes of allergies that we see in dogs, so you get a little bit of an idea as to why itchy skin can be so hard to diagnose and treat. Typing about skin issues can sometimes take a while, so I'm going to start with the allergies, then I'll follow up with other possible causes and testing that can be done.
Since back-and-forth discussion is important, if you have follow-up questions, please get back via "reply", and I'd be happy to address them!
On allergies, these are the main types that cause skin issues (though we can see other types that cause the classic respiratory issues in humans)
Flea allergies - Dogs can be allergic to fleas, and you will notice inflammation, rashes, and bumps in areas which fleas like to hide, such as bellies, neck and facial areas, along the back, and around the rump. The best way to treat flea allergies is via strict flea prevention (Frontline or Advantage should be applied every 30 days). Year-round treatment for fleas is recommended, as it takes a cold snap of below zero weather for thirty days to completely eradicate the flea population. They can also lie dormant in your carpet or cracks in your flooring, so it may be necessary to flea-bomb if an infestation gets bad.
Your pet does not have to have an infestation of fleas in order to display flea allergies. The allergic response caused by fleas occurs after the flea bites your pet. Since flea preventatives also typically work AFTER a flea bites your pet, then there is that opportunity for a flea to jump on your pet and bite before the flea dies, and that can trigger the allergy. Because of this, it is often good to have a secondary medication on board to help with the itch. Allergy shots, Atopica (which is a slight immune depressor to help modulate the response), plain Benadryl at 1 mg per pound for itchiness can help a lot of dogs, hydroxyzine can be prescribed for animals that are still struggling, even on the Benadryl. All the secondary drugs are designed to ease the itch, not cure it, so the flea preventatives are still important.
Food allergies - This is a fairly common cause of allergies in dogs, can cause many skin issues in a dog similar to what you are seeing, and they can be seen all over the body, though areas around the face, ears, back, and rump tend to be the most common areas to see inflammation. A good hint that your dog is suffering from food allergies is frequent ear infections. Studies have linked frequent yeast and bacterial infections to food allergies, and when the allergy symptoms are treated, the frequency of ear infections also fade. Food trials can be done to help identify which food ingredients cause your dog the most issue.
With food trials, your vet will help introduce your dog to a "hypoallergenic" diet which you should keep to very strictly (no treats or human food should be given while on a food trial) for around 12 weeks to determine if there is any improvement from symptoms while on the diet. You can also try switching foods at home to a diet with ingredients vastly different from the one you are currently feeding (IE, if you are on a chicken and corn diet, try a lamb and rice). Many vets carry prescription diets that have fun ingredients such as kangaroo or venison and green pea to help people find a diet with different protein and grain ingredients for dogs with food allergies. Any time a food is switched, it should be done gradually, mixing the old food in with the new food over a period of nine days (3 days of 75% old/25% new, 3 days of 50/50, and three days of 25% old/50% new) to help prevent against gastric distress that comes with a sudden food change.
While switching to a grain-free diet CAN help, and a lot of people will tout them as a cure-all for skin issues, many dogs are actually allergic to the protein like the chicken, beef, or fish. Finding a novel protein like venison may be the better solution in these cases.
The same secondary medications as above can also be used at home to help control the itching until the skin clears up after the right food is found.
Contact allergies - These are the things a dog touches that cause allergic reactions. They can be a wide variety of things, like carpet, grass, nylon, certain plastics, etc. The best way to treat against this is by identifying the cause of the problem, and then getting rid of it in your dog's environment, (Obviously difficult to do with grass). This is the least common of the allergies. allergy tests can be requested from your vet to give you a specific answer about what your pet is allergic to, though these can be expensive.
The secondary medications are also helpful to control the itch here, especially if it is something your animal has to come into contact with, like grass.
There are many things that an cause allergies, and identifying the cause can be time-consuming and costly going through trial and error. However, if you are having trouble with the medications you are giving at home, sometimes allergy testing can end up being cheaper in the long run.
Some things you can do for your dog with allergies when your meds aren't working:
Talk to your vet about a prescription shampoo. There are shampoos that have hydrocortisone as an ingredient that can help with skin inflammation. There are also shampoos that have fungal medications, shampoos to help with mild skin infections, and playing with a variety of these may help pinpoint the issue too. You can also buy shampoos at the pet stores that contain oatmeal, which can help absorb and rinse any airborne allergens from your dog's fur when you bathe.
Add omega-3 fatty acid supplements to your dog's food. You can get these from a vet, or buy the fish oil pills from a pet store or local grocery store, and puncture and squeeze a pill over your dog's food. Sometimes this can help your dog's allergies, and it's certainly good for their skin. If the current fish oil you've got isn't working, let your vet know, and they may have a different brand they'd like to try.
Sometimes dogs can scratch at areas so much that they develop secondary bacterial infections. You'll notice that these areas become really inflamed, sometimes crusty, and sometimes look "oozy". In cases like this, you will want an antibiotic from your vet to help get rid of the bacterial infection, rather than just treating the allergy itself.
Since your vet thinks this is allergy-related, the above information is probably where you want to start. I am going to go ahead and submit this part of the answer so you can read it while I start going into things aside from allergies that we can see in dogs.