It varies – there is generally an immediate effect, especially on the emotional level (i.e. Peace and Calming can immediately help to relieve tension and stress just by breathing it.) For longer-standing issues, it can take longer to have an effect, and for acute issues it can work right away (like Purification on a bug bite for example, and peppermint can often relieve a headache quickly)
I’d talk to whoever has prescribed it first. And, it makes sense that you might not enjoy the scent of one that is actually needed. Funny, but we don’t always like/enjoy what is needed for healing and transformation! If it’s not specified where on your body to place it, one way to deal with this is to put it on the bottoms of your feet. An excellent way to absorb the benefits, without needing to smell it.
You might want to take a class (a lot of people who sell Young Living or doTerra offer free classes to introduce people to their products, and there are reputable aromatherapy classes online if you’re interested in learning more), or get a book or two to start. If you start looking at bookstores or online, you’ll find dozens of great essential oils books, both reference books and more in depth books about aromatherapy. I also have a blog post with some basics to help you get started, here.
Lavender is fine to wear – safe for skin even for children. Citrus oils need caution on the skin. They can cause a sensitivity/reaction, so you can either dilute with some vegetable oil, like almond or jojoba oil, or diffuse in a room, which is very beneficial as well. If you’re fine with it in your bath, it’s probably spread out on the water enough to be fine. Water doesn’t dilute EOs though, only regular oils, like almond, olive, coconut oils do. If you need to remove an EO, use a vegetable oil. Citrus are generally uplifting and wonderful to use in your home.
If it’s safe for the skin, which most EOs are, it’s fine to put anywhere (not eyes or other tender skin), and remember a drop goes a long way! You can always dilute with a vegetable oil if you’re not sure, and watch for any reaction. There are a few (bergamot for one) that create sun sensitivity, so you need to be sure to put those somewhere covered.