EVE New Citizens Q&A

First, while most ships can carry some drones there are ships with bonuses specifically for drones. These bonuses not only include improved dps and HP's for drones, the drone bonused ships normally have larger than average drone bays to let you carry more drones (you're still limited to using 5 drones max, but you can carry more spare drones in case you lose some or you want to switch to drones with a different type of damage).

Gallente have a T1 ship in every category from frigate to battleship that has specific bonuses for drones. Amarr also have a similar lineup of specialised drone ships. Amarr drone bonused ships tend to have the largest drone bays, but most of the Amarr drone bonused ships have less drone bandwidth than the equivalent Gallente drone bonued ships. There are a few other ships in some of the smaller NPC factions that also have drone bonuses. Caldari and Minmatar don't have ships with bonuses specifically for drones.
http://wiki.eveuniversity.org/Drone_Capable_Ships

The next point to realise is that not all drones are equal. A Minmatar light drone does not have the same stats as a Gallente light drone, same with Amarr and Caldari light drones. Some drones have more HP's than other drones. Some drones are faster than others, Some drones have a longer weapon range than others. There are tradeoffs in this, if you want more of X, you'll have to settle for less of Y.
http://wiki.eveuniversity.org/Drones#Light.2C_medium_and_heavy_drones

You've already been told about the Drone Durability skill, but there's also a Drone Durability Enhancer Rig you can fit to your ship which will give your drones some more HP's (You'll need the Drone Rigging skill at 1 to use the T1 version of this rig). There are versions of this rig in every size category to suit frigates all the way up to battleships, and getting the Drone Rigging skill to 1 will unlock the T1 rig for every ship size. You'll need to train the skill to 5 to unlock the T2 version of this rig.

There is one more thing you might consider doing to kelp keep your drones alive. Fit a remote shield or armor repairer to your ship, or carry a shield or armor repair drone (repair drones have their own specific skill needed to use them). You'll have to use a high slot on your ship to fit the repairer, and when you see one of your drones taking damage, you'll need to target that drone just like you would target a hostile ship, then use the remote repair module on the drone.

The down sides of doing this:
You have to give up a weapon or some other high slot module to fit the remote repair module.
Remote repair modules have a limited range, you'll have to keep your drones close to your ship to use it.
If your capacitor use is already tight, this will make it worse.

Using a repair drone doesn't tie up a high slot, has more range flexibility and doesn't drain your capacitor, but you have to give up one combat drone in order to have that repair drone in space (less drone dps), and repair drones repair much less than a remote repair module fitted to a ship (a single repair drone will have trouble keeping up with the incoming damage to a combat drone).

Using a remote repairer or using a repair drone both require you to target your own drone which means you can target one less hostile ship, and you might shoot your own drone if you are not careful.

The Gallente rookie ship the Velator not only has a bonus to drone hitpoints and damage but also has 10 m3 bay and bandwidth so you can field 2 small drones along with the bonus. In that resepect it is like a double bonus. The Ibis does have both a hybrid and missile bonus but still only has 2 high slots.

I've never run EFT numbers on the rookie ships to see how that all works out but I thought it was interesting in this context.

^^This.

It's a fundamental decision you have to make whether to micro-manage expensive drones or 'launch-and-forget' cheap drones. I'm mostly a micro-manager...selecting targets for my drones...grouping different drone types...recalling and deploying repeatedly...etc.

But there have been times when I've let a swarm of T1s handle the fight on their own and then replaced them as they expire.

The thing to remember is that drones are (generally) dumb and have a tendency to go looking for a fight if you let them. Often a fight they cannot win.

Once upon a time Rats aggroed and then kept that aggro until their demise. Of COURSE that was abused by the players by drawing aggro on themselves then releasing their Drones and going AFK (the Rattlesnake was the favoured build for that).
CCP eventually countered that... and overdoing it, as usual. Drones got murdered left and right and the forums were full of delicious, sweet salty tears. After a couple of fixes we are were we are now.
So much for history.

As said above, there is really no way around watching and micromanaging your Drones, however in the long run I would advise getting around that problem by simply reducing drone usage to a minimum simply out of practical sense: Read up mission guides to see which Frigs can scram you and kill them while they are incoming, otherwise just ignore them; as long as they do not web or scam you they are harmless. Also, a Micro Jump Drive apparently works wonders.

OK the part about not being able to repair your drones in space is 100% false.
You can use remote shield, armor and hull reps on your drones in space.
You can use shield and armor bots on your drones in space.
Bot are simply drones that boost shields or repair armor instead of shooting things.