My question is, is there any way ("which makes sense") to use my Tamron SP 70-300mm F/4-5.6 lens for Nikon with this Olympus OM-D E-10?
Not really. While there are Nikon F → micro four-thirds adapter rings, the fact that your 70-300 doesn't have an aperture ring on it means that you have no way to control the aperture on the lens. You also lose autofocus capability, wide-open metering, and EXIF information from the lens. If you're planning on using this lens to shoot wildlife or sports, the lack of autofocus alone makes it a non-starter.
See also: Can I use lens brand X on interchangeable lens camera brand Y?
...Or should I better spend more money for a bigger standard lens like 14-150mm?
Superzoom lenses are great for focal length versatility, but often suffer from some form of image quality compromise to cover the large zoom range. I'd actually say that you might be better off pairing your 14-42 kit lens with a dedicated telephoto zoom (e.g., the Panasonic 45-200 or the Olympus 40-150). It'll cost less, and the image quality will probably be better, especially at the telephoto end of the range.