Red is actually not a trendy color. It’s not because it is not “in”, but because this basic color is never really out. Even if it’s not your favorite color, you should know that there are a lot of people, who like it. However, we don’t see red sites that often. Orange, on the contrary, is something trendy. Not as favored by the masses as red, it is nowadays often used by somebody, considering himself "cool".
Red background is very emotional, aggressive. Combination with black amplifies its impact. If you add bold sans-serif typo, you will get a typical propaganda impression. Drawing attention this way may be seen as annoying.
Red text over white or black background would rather be ignored as a typical advertisement, than attentively read. Primary red, especially combined with white backgrounds and other bright colors, might also achieve a bit cheap impression. If that is your goal – fine. If you want to have more noble touch, you might experiment with darkening the red tone to bordeaux or leather red-brown. Dark red is much calmer than primary red and it makes it suitable as a good background for white or yellow text.
Have you ever seen amateur websites with orange backgrounds? I would bet for “no”. The reason might be that it is disliked by many people and is almost never a favorite color for ordinary people. Orange is something for designers and advertisers, only they may feel comfortable with it. Some modern agencies even choose it as their corporate color.
Orange is bragging, extravert and artificial. An orange site would suggest openness, happiness, vitality, innovation and, most of all designer’s courage.
Bright red, orange and yellow can be well combined, creating a warm, positive mood. Adding white to this combination would create a more formal look. Often orange is combined with grey.
Darkening orange and yellow is much more trickier than red, because you won’t get “dark orange” or “dark yellow”, but rather a completely different color instead (mixing black to orange would make brown and mixing black to yellow would make, well... quite a wide palette of colors, which you would doubtfully like to use in most cases).
The combination, which I consider highly critical, is red and green. It is especially irritating when typography is concerned. It is almost impossible to read a longer passage of green on red or red on green.
I would also not recommend using the combination of yellowish red/orange (cadmium red/orange) with greenish tones, unless you want to implicate, that the design comes from India.
Here are a few sample palettes, suggesting combinations of background and foreground (i.e. fonts) colors: