Most people find their heart & lungs react well to early training. As Sussex Runner says, the legs have some catching up to do. The danger can be if you listen to your heart & lungs, and make them push your legs too far too fast. That's a great way to pick up a niggly injury!
Patience is the key. Follow a recognised programme. Throwing in some low-impact leg work, such as cycling or gym work can help too.
For running, make sure you check your shoes are OK. Sportsdirect might claim to be cheap... but are they're not as cheap as they portray... and are utterly hopeless with advice - which I consider to be worth a few quid.