The work is already underway. It was thought to be virtually impossible because in all the samples of mammoth DNA there were way too many gaps. They died out well over 100,000 years ago and that's a very long time to naturally preserve tissue.
The best we could have expected was to fill in the gaps with an existing species. However, the discovery of a very well preserved mammoth bone 18 months ago has changed all that.
For me, it's first off an ethical question. There is no mother mammoth to carry the clone so we'll have to use a surrogate (probably an elephant). Subjecting a creature to carrying another (not of its species) is one thing but there's every chance of permanent damage to the elephant.
If the mammoth is delivered, it will be into a world where it is the only living example of its species. We don't have another mammoth sample to create mammoths from so there is no diversity and the only other mammoths it could interact with are clones of itself. It would be genetically too narrow for a breeding programme.
You asked about the bigger picture, though. I doubt any cloned and previously extinct animal would be released into the wild. They'd be too important to science and would spend a lifetime being studied. Also the chances of finding DNA we can work with is so slim that it's only happened once (albeit recently). The only creatures we could bring back are those that have been frozen since death (meaning they had to live in a frozen habitat that stayed frozen). I'm surprised we're able to do this with a mammoth. A sabre-toothed cat is much more likely (they became extinct much more recently).
Beyond the big animals, I'm guessing the only possibilities are insects, spiders and scorpions trapped in amber. It would be hugely irresponsible to introduce an insect from the past and could potentially have dire consequences for life on this planet.
