Your friend is wrong. You don't have to get a 55-something telephoto zoom, unless you don't want a gap in focal length coverage. A lot of us would say that the 55-75mm range probably doesn't matter, while the additional length of a 300mm lens over 200mm lens is probably worth it. If you do care about range coverage without a gap, then getting an 18-300 supertelephoto might be a better alternative, but with more image quality compromises at the ends of the range.

You can use FX lenses on a DX body with no issues. And, as long as it's got AF-S, it will autofocus on a D3x00 entry-level body.

However. You do need to be aware that using a telephoto zoom lens can be more difficult than using a simple 18-55 walkaround. Most reasonable-cost 70-300ish telephoto lenses tend to be slow (i.e., have a maximum aperture in the f/4.5-5.6 range) to keep their size small and the cost low; and like any lenses, will typically perform better stopped down from wide open (i.e., f/8-f/11). 300mm is a lot of a reach, and a lot of magnification which will effectively increase blur from camera shake while handholding. Know the 1/focal_length rule (i.e., that you want to use at least 1/300s shutter speed for a 300mm lens; or 1/450s if you count the "crop factor" of a DX camera). Know good long lens techniques, and consider support gear if you need slower shutter speeds.