Work experience is the most important aspect, and lots of it.
Many applicants will be reapplying and will have an additional year to gain experience so it is not uncommon to be up against applicants that have gained 20+ weeks experience. While it is true that you do not need to have this much, the additional insight of the profession gained from this is what counts.
I would recommend seeing practice within a small animal veterinary practice and large/equine if possible also for a minimum of a week in each. Even seeing multiple different small animal practices can show beneficial as just because something is standard practice at one does not mean so at the other.
On top of this, and arguably more crucial, is animal husbandry experience. You won't realistically be able to do much at a vets but stand around and watch as you lack the skills/knowledge to be of much assistance but you can get stuck in with husbandry. As well as looking good on your application this benefits you by providing you with first hand experience of handling and caring for animals in different environments. And once again visiting another similar establishment is good for comparing standards and techniques.
I would say must haves animal husbandry wise are:
Lambing
Dairy
Dogs/cats - kennels and catteries or animal shelters
Equine - riding schools are a good place
These are pretty much core placements though if you can visit beef farms and piggeries this would be highly beneficial too.
Then there are bonus placements that you don't need to get an interview but can make you stand out on paper and in interview:
Abattoirs - would highly recommend, a large part of the profession is public health and on the course you will have to do compulsory public health placements
Veterinary laboratories - again - disease control/public health is a major part of the industry
Zoo's
As far as additional stuff, I never did it and know plenty on the course that both do and don't have Duke of Edinburgh. If its something you enjoy and feel you have time for then go for it!
Personally I was working before I applied and a lot of my interviews were focussed on my job (kennel supervisor) and how I dealt with situations/people.
As said in the post above think about the qualities that make a good vet, and how you can demonstrate you possess them