I have a Tamron Minolta AF 28-300mm f/3.5-6.3 LD aspherical macro lens, which I would like to use on a Canon EOS 600D or Nikon D5200 DSLR. The lens was used on a film SLR Minolta camera. Is there an adapter which will make this work? I want to purchase a new camera and would like to know whether this lens can be used on another camera.
Consider Sony A-mount
Your Tamron lens is Minolta AF mount. This is, in fact, the same lens mount that Sony's A-mount SLT cameras use, so if it's absolutely imperative that you use this lens, Sony's A-mount cameras would not require an adapter and would give the highest level of compatibility (including autofocus).
However, to most of us, a third-party slow superzoom might not be worth giving up the higher flexibility and ubiquity of the Big Two makers (Canon and Nikon). Sony seems to be dedicating more of its development resources at the moment to the mirrorless E-mount (NEX) cameras and lenses than to the older A-mount; and Sony A-mount high-end lenses can be more costly than high-end lenses in the Canikon mounts because they're Zeiss. In addition, note that Tamron currently makes a US$200 18-200mm f/3.5-6.3 lens for crop bodies in both the Canon and Nikon mounts which would do most of what your 28-300 did for you on Minolta AF, and would, in addition, actually be wide angle on a crop body.
Flange Focal Distance
Adapting a Minolta AF lens to Nikon or Canon is not something most folks would do. The key consideration here is what is known as the registration or flange focal distance. This is the distance from the image plane that the lens is held by the lens mount. Each mount system tends to have a different distance. Adapting a lens from a deeper mount to a shallower one is easy--you make up the distance by adding a hollow tube--which would be required anyway, since the physical coupling differences also need to be accounted for. But adapting from a shallower mount to a deeper one is problematic, because you can't just shove the lens farther back into the camera, and lens or camera mount modification will only work to certain tolerances.
When a lens is held farther away from the image plane than it's designed to be held, its ability to focus is limited. Think of it like using macro extension tubes. You increase the ability to focus close, but you lose the ability to focus past a few feet to infinity. While this might work for portraits or macro, it severely limits the usability of the lens. And the only way to regain the ability to focus to infinity is to put a glass element in the adapter tube so the adapter acts like a short teleconverter.
Adapters with Glass Elements
Teleconverters, however, increase focal length, reduce maximum aperture, and typically add softness. The cheaper the glass element, the more it's liable to compromise image quality. If you are on a very low budget, and image quality is not of paramount importance, or you are adapting a very valuable high-quality lens, such as a macro lens, this may matter less. But many people who do adapt lenses eschew adapters with glass elements in them.
Unfortunately, the Minolta AF mount flange distance is 44.5mm. Nikon's is 46.5, and Canon EOS is 44mm. So, the Nikon mount is deeper, and the Canon mount, while shallower is only shallower by half a millimeter, which isn't thick enough to make an adapter ring. So any adapter rings you will find will have glass elements in them.
Additional Adapting Issues
You also need to understand that when adapting lenses you lose electronic communication between the body and the lens. No EXIF information from the lens. No way to adjust the aperture of the lens from the body (i.e., you have to use stop-down metering, and can't use any automated modes that rely on adjusting the aperture, such as shutter priority). In addition, the Minolta AF lenses do not have aperture rings, so without this communication, you have no aperture control at all. And, of course, you will lose autofocus.
Adapting lenses is a pain. And while it may nominally look like saving money, in the end the inconvenience of not using native autofocusing lenses may look more than worth the higher pricetag in the end, unless you are someone who is luddite enough and stubborn enough to just do something the old fashioned way because you can, or because you have a specific love of collecting vintage glass.