Shipping from UK
First of all, it's important to use a reputable removal company. The best are not cheap but the cheap ones will cost you more in the long run. Go for a firm that is affiliated to global removals groups, particularly affiliates of the Latin American Removers Association.
If you don't have enough stuff for a 20 foot container, use this board to look for others who might be in a similar situation and willing to share. Your container will probably be shipped to one of the ports near Santo Domingo...Caucedo or Rio Haina, there is an infrequent service to Puerto Plata but often the container gets diverted to Haina anyway.
It may be better to reconsider the idea of shipping stuff altogether, unless you have some really special stuff, it might be cheaper to buy here rather than ship from there, pay customs duties and have the hassle and heartache of clearing and inland transporting. Electricals here are nominally 110volt not 220-240volt as they are over there so your fridges, washing machines, etc., are best disposed of there rather than brought here expensively and then you won't be able to use them anyway.
The climate here is not really suited to heavily upholstered furniture, carpets and curtains, many areas are extremely humid and not really good for veneered woods, chipboard and MDF whereas reasonably good custom made furniture in resilient local wood can be obtained comparitively cheaply here.
So my two cents worth say, think whether you really need to ship at all. If you do, bring the minimum and look for good movers.
Good Luck!!
Bryan