There are a few different ways that allergy testing in performed. One way is to measure the amount of antibodies in the bloodstream against specific allergens. Another way is skin testing, which involves injecting small amounts of the allergen into the skin and looking for an allergic reaction. Both of these tests are typically performed by an allergy doctor
However, it is generally a really bad idea to go looking for allergies without any specific reason. It is exceedingly unlikely that all of your 'small health problems' are related just to a simple allergy. If you have not talked with your primary care doctor
about this plan, you need to do so to get their opinion about what might be causing your multiple small health problems.
The danger of performing allergy testing without a good reason to do so (meaning, a high suspicion that you a truly allergic to one or more specific allergens) is that the testing is likely going to turn up something that is slightly positive, as these tests are not very specific and are sometimes hard to interpret. Out of context, this positive test might not mean anything.
Start by getting a formal evaluation for your small health problems with your primary care doctor. They will determine if allergy might be an issue and if it might need to be tested for.