What type of Drill should I go for?
First you need to decide if you want one drill to do everything. It isn't always the best choice but it can be a good choice if drilling into brick or concrete is rare.

Functional types


Drill Driver - should include separate settings for drilling and driving. 
There will be a variable torque setting that avoids tearing the heads of screws.
Combi Drill - Adds a hammer-action function for more efficient (faster) drilling
into brick and concrete using masonry drill-bits.


It is worth saying that a separate impact driver is probably a good buy if you do a lot of driving of screws. It also saves you from having to frequently swap between drill-bit  and screwdriver-bit.

You can get hex-drive drill-bits for drilling small holes. They are more expensive but this quick-change ability might be useful if you have only one tool for both drilling and driving.

Power

You also need to choose between corded and non-corded


Battery portable - much more convenient, especially for outdoor use.
Batteries need replacing every few years and are not cheap. 
The drills themselves are not cheap, though budget brands exist.
Corded - Generally much cheaper and more powerful. 
You have the obvious inconvenience of a lead. A Corded drill and an extension lead
will probably be more effective for drilling brick and concrete.


If you pick a battery drill, you'll need to consider voltage and capacity (Amp-Hours or Ah). For lightweight occasional use a 10.8V drill is adequate. The newer models are very good. If you really need more power and don't care so much about weight and handling qualities you could go for a higher voltage system. 

You sometimes get a choice of battery chargers, the more expensive ones monitor battery-state and prevent overcharging. I've ruined a battery by accidentally leaving it on-charge overnight instead of for 3 hours. A plug-in timer-socket can help. More recent battery types (i.e. not NiCd) may be less prone to this issue.

If you buy a corded drill, it is worth considering ones with a standard 43mm collar. This would let you later use it with a drill-stand.

Masonry

If you really want to drill deep, wide holes in masonry as easily as in butter, buy a corded SDS drill. Also good if your main objective is to impress friends and passing females†. I suppose you could use one to drive screws in much the same way you could use it to hammer in nails.

Brand

Note that most manufacturers have both home-owner and tradesman brands (e.g. Bosch have "green" and "blue" ranges, DeWalt is the more tradesman-oriented brand of Black&amp;Decker). Unless you can get a special price or expect to use the tool many hours every day of every week -- you should probably buy a non-professional brand.

There is considerable range of quality in drills. In my opinion, the cheapest stuff is only worth buying if you don't need precision or reliability and expect to write-off the investment and throw it away after a single project. If you want a tool to last ten years I would buy the better-known US, European or Japanese brands - who are likely to care about their reputation. A lot of big-box store-specific brands seem to be custom-badged cheap very low quality products made in anonymous factories in China.

Me

My first drill was a Black &amp; Decker corded two-speed drill. I still have it (it is ancient) but nowadays it sometimes makes a nasty shrill noise on the high-speed range so I replaced it with a Bosch corded combi-drill that was on special offer. I don't use this often

I have a Bosch blue 10.8V combi-drill that has 2Ah batteries. I am very impressed with this. It replaces a heavier Black &amp; Decker 12V combi drill that I broke. This is what I use most.

I have a small Bosch 3.6V screwdriver for convenience (saves swapping bits and is v small) - I wouldn't expect it to drive hundreds of screws in a session!



† It won't. They might humour your self-deception a little though.