There are probably two issues here. Firstly, braces make it difficult to clean the margins of your teeth, this will lead to a build up of plague here and cause gingivitis. Don't worry about this too much, its a reversible condition, no lasting harm will come of it but you do need to improve your brushing. As the person above said the best thing to use is some little brushes e.g. teepee (sp?) brushes or other interdental brushes. There is no need for corsodyl, its for more serious conditions than this and though it would benefit you it could also stain your teeth and can temporarily affect your taste among other things. The other issue is that it sounds like the brackets on your teeth are rubbing against your lip making it sore and bleedy. When I had my braces they gave me some wax to put on the brackets if they were hurting my lip, this could be a solution. Else eventually your lip will toughen up a bit and it won't be an issue anymore.
If you want some proper advice (i.e. not from a trainee dental student) then ask either your orthodontist or dentist, whoevers appointment you have sooner, they will both know about this problem.
Also to Dhesi - toothpaste doesn't strengthen gums, it strengthens teeth due to the fluoride in it. And also gets rid of the bacteria which causes gingivitis - this is why it stops gums bleeding.