The only pure essential oils I own are peppermint, melaleuca, lavender, lime and orange. I would recommend investing in these first if you do not already own any, and this is why:
Meleuca oil (or Tea Tree Oil) is a first aid kit essential and is excellent at healing skin infections--whether they be bacterial, viral or fungal. It is also amazingly effective at healing sunburns overnight.
Orange oil is also an anti-viral and is rich with anti-oxidents and vitamin C. Both orange and lime oil are great first-buys because most everyone enjoys its highly oliferous scent and many associate its aroma with being "clean" or "purified." This is probably because that's exactly what citrus fruits and their corresponding oils do--cleanse and purify our bodies and our surroundings. Both of these oils can also be used in recipes or used to flavor your water if you 1) use stainless steel/glass water bottles/glasses (essential oils leech the chemicals out of plastic) and 2) if the brand you purchase are food grade quality.
Last but not least is my personal favorite: Lavender oil. Lavender is great for cold sores, tense muscles, cellulite, infections, inflammation, sprains, aches, eczema, acne, insect bites, insect repellent and simply inhaling deeply. Lavender is especially known for its calming and stress-releasing properties. I often smell it while I am studying or put a drop under my nose or on my pillow before going to bed. Lavender is easily one of my top favorite oils.
If you are wondering where to buy food grade, therapeutic-quality essential oils I recommend looking up DoTerra. Although their oils are rather expensive (upwards of $15 depending on what you want) they do have excellent quality and seem to be extra potent in comparison to the variety you can find at your local health food store. :) You can also find much cheaper essential oils online or at your hometown Good Earth or Whole Foods, which is where I got my lime and lavender oils for under $20. ♥