Allergies generally feel itchy in the areas affected. These include about everywhere you can imagine, eyes, ears, nose, throat, chest, and any skin surfaces. Allergies can additionally give congestion from swelling and inflammation of mucous membranes that can produce mucous that adds to the misery. This mucous can fill sinuses, and drip down the back of the throat causing such a pain that it's hard to swallow, mimicing infection. Clearing this mucous away with saline nasal rinses helps many remove the cause of suffering. The swelling and mucous can even cause a lingering cough, both from the irritation from the post nasal drip, or inflammation of the bronchi, your breathing tubes.
Rarely, allergies can cause a life threatening reaction called anaphylaxis, where the swelling and inflammation spreads to the throat, and if swollen enough, can block off your airway and you can't breathe. For people susceptible to this condition, it's important to always carry an epi-pen, a self-injecting pen that delivers epinephrine under the skin to decrease that swelling and let you breathe again.
Food allergies are a controversial topic that's argued by different medical specialties. The intolerance to certain foods is sometimes mistakenly called an allergy even though it's due to a lack of an enzyme, such as lactase in the example of "milk allergy". True anaphylaxis to foods is rare, even though many people have variable reactions to nuts ( e.g. peanuts) and seafood that they consider allergies for example. The reason for the difficulty in figuring out whether you've got a food allergy is that traditional testing for airborne allergens doesn't work well for food allergies, and the blood tests are even less helpful. The kinds of reactions to foods are so varied (rash, nausea, diarrhea, bloating, gas) that it's hard to know what actually is happening, and from what cause or process.