You can decide if you want to keep a HW cylinder or not.
If you do it be probably be best to upgrade it to a new fast recovery model, then have a new condensing 'system' or 'open vent' boiler fitted.
You have also got the option of having a combi fitted to have instant hot water on demand. These are very popular these days, but not necessarily perfect for all households, especially those containing more than one bathroom that may be needed to be used at the same time.
One of the biggest factors you will need to decide is where the new boiler will be sited, bearing in mind flue routes and pipe work access etc.
There is also an outside chance that you may new some new radiators, especially as you say your house doesn't feel too warm.
What make of boiler you get will depend a lot on your budget, but as long as you avoid anything made by Ideal and possibly Ariston you won't go far wrong.
Your local installer should be in a position to answer any and all of your questions fully, a lot better than any of us on here, as he will see your individual circumstances for him self.
Remember you have what you want, after proper discussion/consideration, not necessarily what an installer wants to push onto you.