Choosing a rose is somewhat more complicated than simply choosing it for the flower colour. Roses are also not particularly easy care plants - they require deadheading, spraying against pests and diseases, and pruning properly once a year. Most roses prefer as much sun as possible, but there are varieties which do quite well with much less sun on north or east facing walls. If you intend to plant rose in a spot that does get sun if the sun is out (south facing for instance) then you can choose one that likes more sun - whether you're aware of it or not, it will get more sun than one planted against a west or north facing wall.
That said, the other decisions for selection are based around whether you're choosing a climbing rose to grow against a very large, broad wall (such as a house wall), whether you want it to grow against a fence that's only 6 feet high, whether you want it to cover a wide space but not get too tall, or simply grow upright against a narrow wall, fence or pillar,because they all have different growth habits, as well as height and spread.
Often, under 'climbing roses' you may find roses available which are actually rambling roses - these have much laxer, usually longer stems, usually have only one flush of flowers all at once for about 6-8 weeks during summer, and need pruning in September. Most true climbing roses, on the other hand, have stiffer stems and are repeat flowering, meaning they produce blooms throughout the summer - some of these will be quite prolific in their flowering habit, some not so much, meaning you will get flowers, but you may not get an impressive display at any particular time. This depends on variety of rose; 'Compassion' for instance, has fragrant peach coloured flowers, but in my experience usually only has half a dozen blooms fully open at once, whereas climbing Iceberg (white flowers) will have smaller flowers but many more open at once. Both are repeat flowering climbers, and climbers should be pruned in late winter or early spring (probably late March where you are, depending on the winter).
Some roses are are more prone to disease than others too, and some do better in cooler, northerly regions than others do. One good red flowered climbing rose you could consider (if you can find it, you may need to order it online) is one with a very unlovely name - 'Hamburger Phoenix'. The breeder is Kordes, and kordesii roses are known for healthy growth and good disease resistance. Flowers on this one are dark red, slightly fragrant, repeat flowering, and it reaches a height of around 9 feet. It is a good choice for a north or east facing wall, or areas where not much sun is seen. As with most climbers and rose bushes, deadheading regularly is important to ensure successive blooms.
Rosa 'Sympathie' is another Kordes climbing rose with red flowers which also does well on a north facing wall,or in low sun situations. It repeat flowers profusely,the flowers are fragrant,it has good disease resistance, and reaches a height of just over 10 feet and doesn't get too wide. I think Peter Beales online is selling this one.
When you've chosen your rose, if you want to train it against a fence or wall, you will need some kind of support in place prior to planting to tie it onto, so a good solid trellis (not one of those, thin, collapsible ones with diamond shaped holes) or wires fixed into metal eyes on the fence posts or wall, with the heavy duty wire tensioned between them.
Soil preparation is the usual for most planting - dig over the area where you will plant (remember the root ball needs to be a foot away from the wall or fence behind), digging over an area at least twice as big than you need just for the rootball, preferably 3 times as big, add in some composted manure or good garden compost, leave it to settle for a week, then dig your hole and plant the rose, firming in and watering afterwards. Keep the plant well watered during dry spells, especially for the first year or so. After that, most roses are pretty drought resistant. Roses also benefit from a specialist rose food applied twice (April and again 6 weeks later) each year - this is usually granular or a powder which you sprinkle round the base and just lightly turn into the soil. Toprose is a good example.
You may need to spray periodically with an insecticide/fungicide - Rose Clear Ultra is pretty good - some people routinely spray their roses every 2 weeks, but you may not need to do this if you use either of the roses listed here, since they have good disease resistance, although aphids will still colonise young shoots. If you spray, don't spray the open blooms, do spray under and on top of the all the leaves till run off, and preferably spray in the evening when the bees have stopped working on a non windy day.