Looking in the roof, testing the water pumps, checking cupboard space, storage for everything, even dinner plates!
Check the carpets for recently moved furniture - what is it hiding? has it been recently moved - don't be afraid to move and check behind sofa's - check the windows - are they double glazed? When were they last done? Last gas/elec check?
Normally the viewings are done by the agent so don't worry about checking places - you are going to be spending a HUGE amount of money so don't be afraid to ask questions and check.
Definitely use a broker. Get your locations first, prioritise your commute and how far you want to spend getting to and from work and the costs involved.
Make a list of your non negotiables. Driveway or garage. Schools, work, etc.
Then prioritise for the house itself, how many bedrooms, size of lounge, is it future proof, is it liveable? Do you want to host dinner parties - is there room for that and for people to stay?
Have fun - and don't get emotionally involved - be prepared to walk away if you need to.