0603 isn't too bad to solder by hand (I won't do 0402 or smaller though).
SOT23 is probably a good guideline (for diodes too, not just transistors); there are some SOT323s that are smaller that are a pain.
I would avoid certain SOT23-6 parts because it can be very difficult to determine which way the package is supposed to go. (For some dual MOSFET packages it doesn't matter.) We had one where there was a slight bevel along one edge. Grrr.
I would also avoid SOD123 because of the backwards nature, if possible. SMA/SMB/SMC aren't as much of a problem.
And avoid those cylindrical diodes (LL-34 / MELF) like the plague! they will roll off the board.
After doing plenty of research and asking some electrical engineers who use them, I've found some points that were helpful in deciding.
Benchtop desoldering or rework stations tend to have the most accurate temperature controls, best thermal recovery, and extra features like programmable timing and hot-air, etc.
For example, if you plan to remove surface mount IC's on a regular basis, a hot-air rework unit may be ideal. They have nozzles for just about every IC package, and the higher end units can be programmed to pre-heat for a specific number of seconds, then heat for a number of seconds, and some have a vacuum nozzle to pick up the now-free part. (For example, see the Hakko FR-803B, a $1300 unit.) You have to weigh the cost heavily with the amount of desoldering or repair work you expect to do. For the occasional removal of parts, on a hobby level, this is likely overkill.
If you're just looking for a step up (or two) from the soldering iron and vacuum pump or solder braid, self-contained desoldering guns like the Hakko 808 can be found for about $200. While it has less fine control over temperature than a bench unit, it does have several benefits: self-contained vacuum pump, replaceable filters, and highly portable. Unless you have specific requirements for performing repair work, it will probably handle any through-hole work.
There are many online videos and tutorials about surface-mount desoldering. It is more tricky than through-hole for a variety of reasons. If you have to desolder SMT, you will want to check some of them out first. Some irons have tips specifically designed for multi-pin SMD packages, there are also hot tweezers and of course the aforementioned hot-air rework systems.
To recap, here are some solutions for desoldering through-hole and their relative price ranges:
The old desoldering pump, for about $15-20. Works "ok" but requires both hands and sometimes creative angles.
$12 at Radio Shack, requires one-and-a-half hands. I've not used one, so I'm not sure if it would be more or less convenient than a separate iron and pump.
Between $50 and $200. I found some for $50-100 online, but I have actually tried a Hakko 808 and can attest to the performance. Great for through-hole.
$400 and up. Lots of brands to choose from, Metcal, OKI, Hakko, Weller, etc. Once you start looking at stations, a variety of features and use-case scenarios start to emerge. Definitely geared more for commercial and industry use where soldering rework is an everyday occurrence.
One important thing to look for in soldering or desoldering equipment is the availability and price of consumables. Filters, tips and nozzles. They will need replacement. If you find an off-brand version, can you find parts for it later? If you get a quality station for cheap, do the replacement parts cost a fortune? Can they only be ordered from a foreign country?
Many manufacturers provide documentation and videos of desoldering equipment. Take some time to go through them, realistically evaluate what you will use (and what you can afford).
When I started looking, I thought I wanted a dual-port station (one solder, one desolder). Such stations are about $600. I probably am not going to desolder daily. I went instead with a single-port soldering station for $300, and a desoldering gun for $200. For my shop, the portability of the desoldering gun is worth the $100 saved and then some.
A few basic things I would like a hot-air rework station to have:
Other bonus features:
A. "Auto-shutdown" which will automatically powerdown the unit if you place the gun back into the holder.
B. Some units have a cooldown before shutdown feature which will turn the heater element off but leave the fan on until the temperature drops below a certain temperature. I believe the theory is that this will extend the life of the heating element.
You can check out Dave Jone's EEVlog #167 for a review of the Atten 858D+. He explains the basics of what a usable gun should have.