A red color palette is exciting and challenging: Red is an 'advancing' color that attracts attention and can easily dominate a room.
In addition to red color combinations, this page has 3 easy strategies for successful interior design color schemes with red. (More great color combinations here!)

Old Glory blue + white + Old Glory red + unbleached linen (écru)
+ rose madder + navy blue + smoke blue (+ white/linen)
↑ The secret with red, white and blue color schemes is to vary the colors slightly and surround them with natural greys and browns. You can achieve a 'tinted' look for the flag colors by using small-scale, blue-and-white or red-and-white patterns instead of relying exclusively on solid colors.
Best wall colors: White, pale blue, écru.
Best wood stain: Any.

Dark neutral grey + light grey + raspberry ice cream + blue-grey
+ deep cherry red + dove blue + sky blue + cranberry red + grey
↑ If you like a red, white and blue color palette but don't want to be limited to flag colors, do as the French do and just combine any blue & red hues that look good together, then use neutral colors as a backdrop.
Best wall colors: White, light grey, or a tint of one of your chosen colors.
Best wood stain for this red color palette: Rustic oak.

Ruby red + aqua blue + white + grey
↑ Ruby red/aqua blue is a highly popular wedding color palette. The warm, attractive ruby color is balanced by a cool blue that borders on turquoise. This is a delicious red color palette, but it doesn't really tolerate any further hues. You'd have to surround it with a lot of white & neutrals.
Best wall colors: White, pale grey.
Best wood stains: Dark hardwoods.

Coral red + tiffany blue + white + grey
↑ Along the same lines as the ruby/aqua color scheme above, coral & tiffany blue combine a warm, striking red with a cool but equally striking blue. And just like the former, this is a favorite red color palette for weddings.
Best wall colors: White, pale grey.
Best wood stain: Bamboo.

Sage + dusty plum + pomegranate red + duck egg blue
+dusty turquoise + burnt orange + bay leaf green
↑ You don't need a lot of red in a red color palette. Here, surrounded by aged, dusty 'vintage' hues, a rich red hue takes center stage. The burnt orange is essential in this room color scheme - otherwise it would look too dark.
Best wall colors: Sage, duck-egg blue.
Best wood stains: Teak, oiled oak.

Pine + cinnamon + sage green + scarlet + Prussian blue
↑ The hues of a vintage Amish quilt: a red color palette that manages to be simultaneously bold and serene. It would be perfectly at home in a rustic chalet or log cabin. The complementary reds and greens are joined by a very unobtrusive dark blue. (Does 'cinnamon' count as 'red'? I think it does - it's a tone that sits somewhere between true red and reddish brown.)
Best wall colors: Pale sage green or a chalky white.
Best wood stains: Rustic pine, antique cherry.

Tomato red + cerulean blue + pacific blue + pale olive green + light neutrals
↑ Any interior color combination that involves large amounts of tomato red is, of necessity, completely in-your-face. You can lower the impact by choosing soft, greyed tones rather than fully saturate colors. Use the red in small proportion, mainly for accents.
Best wall colors: White, light grey, pale olive.
Best wood stain: Antique oak.

Mustard yellow + turquoise + burnt orange + verdigris + deep berry red
An upbeat, life-affirming 60s red color palette ;-) Lovely for a summer house, garden, back yard, veranda or deck. Also nice for giftwrapping, kitchen accessories ... anything really. Enjoy!
Best wall color: Pale mustard.
Best wood stains: Oiled oak or teak.
Here's some color scheme inspiration to buy:
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