I find that it's pretty easy to do myself. I do tend to find that the color is darker than I expect based on the box description. I'll never forget the time I used "dark brown" and my hair came out jet black! So, with the color you picked, I would expect your color to be darker than your natural color - you say your hair is light to medium brown, and you chose medium brown color. If you are ok with going darker, then you picked the right color. If you want to match your hair color, you probably should have picked one shade lighter (a light brown, and if you want to keep your reddish tint, then a light warm brown (warm means a reddish tint, cool means no red tint)).
Two tips: Don't try to make up for picking too dark of a color by shortening the time you put the color on. 1) it won't work, because most of your hair will pick up the color quickly and 2) it won't cover your grays if you don't leave it on long enough, which defeats the entire purpose of your coloring. With this in mind, I usually pick a shade lighter than I think I need, then leave it on for the longest "cover your grays" time listed on the box.
The second tip is to avoid getting dye on your skin at your hairline, on your ears, etc, is to put a coating of your conditioner all along your hairline and all over your ears. It acts as a barrier to minimize the amount of color that touches your skin, but still easily washes off in the shower when you are done.
It would be easier to give advice if you say what went wrong the first time, so we can help you avoid that again. Update if you have a chance.