I started using oil as a skincare product about four or five years ago. My absolute favorite thing about using oil is that it is the most universal, lovely, nourishing product for the skin and it's very easy to apply, so just about anyone can add an oil to their skincare routine (even folks who are really short on time). My skin type is: (super!) sensitive, redness prone, acne prone, combination (with a tendency to be oilier in the summer and dryer in the winter), with a couple broken capillaries mixed in. When I look for an oil, my number one priority is that it's natural (truly, honestly natural). I use EWG's Skin Deep and Think Dirty to help determine which brands meet my criteria. My number two priority is that the product is gentle and won't react with my sensitive skin. And my final priority is that the product is simple.
As a rule of thumb, I always try to stick with products that are gentle (rather than stringent), even when I'm experiencing a breakout. I find that my skin reacts much better to an infusion of goodness and nourishment to help along the healing process, instead of a product that strips and dries.
For a day time oil, I really like to stick to a one-ingredient oil (instead of a blend). My absolute favorite is Argan oil. It has no scent, it doesn't feel heavy on the skin, and it's light enough to apply well under makeup or on its own. I typically use two or three drops (depending on how dehydrated my skin is) and blend gently with my fingertips. It leaves a lovely glow on the skin that fades as the day goes on. Argan oil is also incredibly affordable, which makes it even more appealing.