This sounds like a fun little project. Let's see what we can do.
A GM Mr. Fix It 2071 ($6,000, Runners Companion p152) has plug-n-play toolkits that include Automotive by default, among a few others. It would seem you could get other toolkits for it by buying them. It also comes with one Auto-soft of your choice at rating 3. It's got Pilot and Device rating of 3 too. Note that it doesn't have rigging capabilities by default.
Let's go ahead and make some upgrades. I'll price this whole Drone out for Automotive as the example, but I'll also include the price for upgrading it's abilities for other applicable skills.
Basic Automotive: First of all, we are going to upgrade the drone's Response. We can't push it up to Rating 6 without installing modular electronics first (SR4a Errata) so that's $1000 and 1 slot already. Response Chip 6: $8,000, availability 16. We should also go ahead and improve the Pilot Program, which counts as a special System software. With response 6 in the drone we can push that up to Pilot 6. That's $12,500 at availability 15, and 0 slots. Next I recommend taking Fuzzy Logic. Fuzzy Logic aids drones in decision making where time isn't a big factor, and we can assume that includes repairs because the Lockhead Vulcan aircraft repair drone (Arsenal) has it built-in for aircraft maintenance. On long tasks, like repairs or building things, it probably gains the full +3 bonus from the fuzzy logic hardware, so it's a worthwhile investment at just $2,500, and 1 slot. Ok, let's get some software. Autosofts are $4000 each at max ratting 4. That's one rating higher then the built in autosoft. Next up we need tools. The Drone comes with an Automotive toolkit already.
At this point you have a Drone able to work on wheeled vehicles/drones and similar. It has a Pilot 6, +4 Autosoft, +3 Fuzzy Logic = 13 dicepool before considering tool requirements/options.
Total Price Tag: $34,000, Availability 16 (2 of 4 slots used)
Adding Technical Skills: You can easily expand this drone into other skills. Aeronautics Mechanic for flying drones, Armorer for weapons and armor, Chemistry for drugs and toxins, Demolitions for explosives, Hardware for electronics/links/etc, Industrial Mechanic, Nautical Mechanic, etc. Each upgrade, including plug-n-play tools, and autosoft is another $4,500, but doesn't use up any more slots on the drone. Don't worry about getting more skills than the pilot is able to handle at once (6) because it doesn't need to have all the skills loaded at the same time. It will just need to have the correct autosofts for any given task, and with fuzzy logic and pilot 6, it shouldn't have too much trouble loading those skills on it's own, or you can simply alter it's load-out yourself.
Upgrade cost per skill: $4,500
Autodoc Plus: If you want it to handle medical situations you need to tweak things a bit more, but it's still cheaper than buying a new medical drone and kitting it out for Pilot 6! Start with getting it a Valkyrie Module, which auto-stablizes anyone put into it, and is expressly able to handle surgery. That's $2,000 and the drone's last 2 slots. It also includes Medkit rating 4. You'll also need some Autosofts. First Aid, Medicine, and Cybertechnology all at rating 4 is another $12,000. If it can't handle the medical situation itself, the valkery module is able to accept telepresence medical assistance from any street doc you happen to have as a contact too (they take a -2 vr surgery penalty, but they can also gain VR and even Control Rig boosts to their skill, according got Augmentation). It's dicepool for First Aid is 14 (Pilot+Autosoft+Medkit) and it's at 13 (Pilot +autosoft +fuzzy Logic) for cybertech and medicine.
Upgrade Costs: $14,000
If you group makes regular use of animals (for magic, biodrones, etc) you'll also want to get a VetTech Auotsoft for another $4,000.
Better Tools: Of course you can benefit from improved tools on almost all the above. A shop is a mere $5,000 for most skills, and may actually be required for some uses of skills, and may provide a bonus on situations where it doesn't. A medical shop is $10,000, but counts as a rating 8 medical "kit."
More Drones: As noted by other posters, you can also have multiple drones using teamwork rules (SR4A p65), but remember you need Autosofts and tools for each extra drone. That's an extra $4,500 for each drone for each non-medical skill. Extra medical drones could be a bit more complicated. Consider getting Evo Orderlys instead of Mr. Handy drones if you decide you want drone teamwork for that.
Black Market Software: Software piracy actually makes this concept a lot more cost effective.
Base price for the drone with black market cracked software: $8,650, with a patch price of $350 every two months.
Each add-on technical skill becomes $900, with a Patch price increase of $100 every two months.
The medical upgrade becomes $3,200 and increases the patch price by $300.
Adding VetTech to that is $400, and another $100 to the patch price price.
As an added advantage to the hacked software option, if you buy more (basic off the shelf) Mr. Handy Drones, you can share the cracked autosofts between all of them (capped at 3 by their pilot/response), so you'd only need buy more toolkits, not autosofts, for each of them to be able to use teamwork rules on any given technical skill.
