First of all, it would be best to find a riding school for your daughter to take lessons at. One affliated to the pony club is good, or one affliated to the BHS (british horse society) www.bhs.org.uk/enjoy-riding/find-a-place-to-ride should guarentee a reasonable level of tuition and animal welfare. Pony club is traditionally for children with their own ponies, or one on loan/as a share, but more centres now are offering membership for those without horses. However, at this stage, I wouldn't worry about pony club too much.
5 is fairly young to start riding, and some riding schools now will not offer lessons until 8. Unless the riding school recommends differently, I would start her off with just 1/2 an hour at a time, to allow her muscles to get used to riding. Riding uses a lot of muscles in ways we don't otherwise use them, so expect her to ache the next day. Riding also requires concentration and can be fairly intensive exercise, so longer than 1/2 an hour can be a bit much for younger children to start with.
To start off, your DD will very likely be on the lead rein or lunge, which means someone else is controlling the pony, and she learns how to stay on, slowly developing more independence. Lessons for young children should be fun though- expect lots of games to be played.
WRT to costs, it doesn't have to be expensive or all consuming unless you let it. You can start off with one lesson a week, which you should be able to hire equiptment for, and take it from there. As she gets older, and wants her own equiptment (e.g. hat, boots, jodpuhrs) things can get a bit more expensive. I wouldn't worry about competitions yet, except maybe riding school gymkanas/shows, which will be very cheap/free. As she gets older, you may be able to drop her off at the yard, so she can learn about stable management (looking after the horses) and help out for the day.
However, buying a pony is a massive commitment in terms of both time and finances, and you do need to be knowledgable and confident to deal with them. I wouldn't worry about this until you are sure you enjoy it and are sure this is something you as a family want to commit to.
I hope you find somewhere for your daughter to have lessons, and she really enjoys it :)