Hi Tom!
I usually avoid using ministers at all. Well actually, I never use ministers, I just don't trust their decissions.
Exploring early on is a good idea, though I usually only explore two systems away in each direction at first. I then go into hyper-colonization mode with some offensive ship building peppered in. Why? Because if I bump into other races right away, they are likely to declare war, but if I wait for contact, it is likely that the aliens will all be tied up in war, and will befriend me... well that's what I think happens
Colonies are life. These are your resources for
everything you want to do in this game, so grab up as many planets as you can and build up those colonies! Try trading with alien races for the other colony technologies (befriend anyone that colonizes a different planet type. this may cost a few resources, but it is definitely worth it)
As to technologies, I think that ship size and ship movement are the two most important areas to research, followed by general weapons (just missiles and DUCs (depleted uranium cannons) for the first 20 turns. Then exotic weapons can become more meaningful.
If you don't have shields, the enemy will likely take out a component every shot. If you have shields, you can probably buy yourself a few damage-free combat turns.
Hope that helps a bit for the start of your SE:IV career.