Don't follow the advice to "cut it down in the spring" if you want flowers! There are three different groups of clematis, which flower at different times of the year and need pruning in different ways. That pruning method is WRONG for the one that you have.
All clematis will grow and flower happily with no pruning at all, but that will result in a "top heavy" plant with all the growth and flowers at the top, which is not ideal.
If your plant starts flowering in May, the flowers are on the previous year's growth. If you prune it heavily in spring, you have cut off all the flower buds!
The main pruning should be done as soon as the first flush of flowers is over (probably in June or July). Cut back the stems to about half their length. You will probably then get another crop of flowers on the old stems later in the year, and there will also be new growth for next year's flowers.
If there is any damage to the plant over winter, you can tidy it up in spring before it starts to grow, but remember that any pruning you do then will be removing flowers.
It isn't clear from your pictures what this is climbing on. A single clematis will easily cover an area 10 feet high and 6 or 8 feet wide in a few years, so it would be a shame to restrict it to a single pole. If you want to move it to climb on a wall, you can get plastic netting to fix to the wall and support it. Another good idea is to let it climb up another plant, for example a tree or a climbing rose where you already have wires etc on a wall to support the rose. Since the clematis stems are thin and light, it won't harm the plant it is climbing on.
If you want to move it, cut it down in winter to two or three stems about two feet long, and then transplant it in spring just before it starts to grow. You won't get any flowers in the first year, but it will be fine in the following years.
There are hundreds of different named clematis hybrids, so apart from a few well-known ones it is going to be hard to put a name you yours. FWIW I don't think it is Nellie Moser - that is pink and white, and has narrower petals than yours.