As an amateur you most likely already have purchased or are going to purchase a tele-zoom lens.
The most common telephoto lenses are the most affordable 75-300mm f4.5-5.6 ones. They often come in a bundle with the body and a standard kit zoom lens. They are not very expensive, especially not lenses manufactured by Sigma or Tamron. I can highly recommend both as an alternative to Canon or Sonys standard telezoom lenses.
You often hear that if you're going to shoot sport or other fast action you will need a professional fast lens, for example a 70-200mm f2.8 lens. The wider aperature (lower f-number) serves the camera with a bigger intake of light, wich also means that the shutterspeed can be a lot faster. On a sunny day or cloudy mid-day you wont need this since the shutterspeed probably will be 1/500 sec. at f/5 and that is more than enough if you are shooting motorsport or soccer. But if you notice any motion blur in your pictures, just crank up the ISO, Most modern cameras will withstand a fair quality and low noise even at ISO 800 or higher.
The optical quality in these cheaper lenses are often very good, especially at f/8 and between 100-200mm.
Such a lens also do good as a portrait lens, at f/5 and 180mm you will get that nice blurred background but still have great sharpness at the subject.
The price of a cheap telephoto lens could be anything from $30 on ebay, or $200 brand new.
A fairly cheap f/2.8 telephoto lens costs about $1200. Before you decide, think trough what youre going to shoot and when. Of course if you're a indoor sport photographer you should consider getting the faster lens.