There is little difference between the interest rate on new fixed-rate and variable rate mortgages and, as the base rate is almost as low as it can go, this means new fixed rates are much more attractive than tracker or discount rates, as a fixed rate will protect against rate rises.
Longer-term fixed rates - five to 10 years - offer the best value as two-year fixes only provide protection against interest rate rises for a period when the base rate is unlikely to rise, whereas the further ahead one looks, the more difficult it is to forecast how much, and how quickly, it will rise.
One caveat is most fixed rates have early repayment charges for the same term as the fixed rate and so it is probably wise to avoid any mortgage where early repayment charges extend beyond the time you are confident of remaining in your current property.
There are two ways to get a new fixed rate - a remortgage, as you suggest, or a product transfer. The latter means you keep your mortgage with the same lender but instead of paying its SVR, which is the rate most mortgages revert to after the initial deal, it offers you another fixed rate. This will probably involve paying another arrangement fee, but you won't have to pay any legal or valuation fees.
Before looking at your options for remortgaging, you should ask your current lender what rates it offers for a product transfer. I recommend you then consult a good independent mortgage adviser to find out the best remortgage options. You can then compare these with what your current lender is prepared to offer. On a remortgage, most lenders offer a free valuation and free legal fees.
In most cases, a remortgage will be cheaper but a product transfer will probably be simpler, providing you don't want to increase the size of the mortgage. This is because for a product transfer most lenders won't require you to provide any new information, such as proof of income.