I'm a hairstylist. One mistake with at home hair colors is that they can be unpredictable, especially on gray hair, since you don't know the exact formula you need. People tend to make the mistake of going to ashy or too gold. Ideally, you want something with a natural/neutral tone and a hint of gold to replace the red color molecules naturally found in the hair. There are some really excellent at home gray coverage colors out there though.
If the red is mostly in the scalp area, that is commonly known as hot roots, and often occurs when the product has been left on too long. If the red is throughout the head, and more of an auburn brown, the red will likely fade bit in a week or so to a more natural golden brown color, depending on the evenness of the color.
You can go to Sally's and get a non- bleaching hair color remover, they are typically extremely easy, effective and very safe to use on your hair. Removers are best used for fresh, deposit only color jobs that went horribly wrong. Since you just colored your hair, the color removal could be a safe bet, so long as it is one that isn't a bleaching method. If you are going to fix it at home, go to Sally's and get the removal, or wait the week and ask the stylist to fix it, you don't want to put another color on top and possibly crate a more difficult disaster to fix.
All medium brown hair has red in it. The color has both blue, red and yellow pigments. So the reddish tint is part of your natural color. If the reddish is only after you color your hair, it is likely because you are choosing a color with too much red or gold in it. So choosing a color with a hint of ash is okay, I would more so recommend using a color with a N/neutral base since too much ash can make your hair turn greenish or even 'muddy' looking, and can make you ends are pull too dark. If you go to Sally's you can choose a color that has natural and ash, such that will look like a 6/NA or 6/01 or something along those lines. Ask the ladies at Sally's for a color formula since they can see your hair. Use 10 or 20 volume developer depending on the color you go with, again, ask for help there.
Per the colors' timing instructions, paint the color on your scalp/roots and leave on for the duration of time, minus approx 15 minutes. Then, very slightly spritz your hair with water and comb it through. Take your excess color, and color the rest of your hair, to about the last inch or so of your ends. Leave on until the last 5 minutes, then comb that color through to your ends. It will help not get that muddy/too dark end color. The reason the ends are going too dark, is because the hair is older, has more damage and is more porous.
Or, you can go to a salon (a better choice really) and have it professionally color corrected, they can then give you tips on how to do maintenance touch-ups.