There are several different ways that this can be accomplished. One major consideration is that stripping and refinishing molding in place can be a very messy process. Removing the moldings and stripping and refinishing can also be a very messy process, but I do have 2 suggestions. One is to hire a professional finisher who can do things like shoot tone spray over the current molding and wainscotting and then apply an overcoat of poly if this will be appropriate for your wood or woodlike materials. This really isn't difficult but there is a level of artisanship needed and it very likely will be cheater than if you do it yourself and have any problems. Another approach would be to paint the wainscotting only with an appropriate color that will make it all blend with the rest of the room. This would be more DIY friendly a project.
If you do elect to check out local professionals, you can start by asking at a local paint store or the store for one of the big paint manufacturers, like Sherwin-Williams. The people who supply the pros generally know who is good at doing what your job requires.