I have not seen this specific questions asked in the past, though there are posts about how to use the four-speed auto gearbox in a manual shift manner.
Let me dispel a statement you made up front: the transmission as found in the E500E is in no way an "exciting" transmission and provides a competent, adequate and functional experience. It gets the job done with no drama or hassles, but particularly if you are used to modern "8-speed" type transmissions, or manual transmissions in sports sedans, the E500E transmission is pretty much the antithesis to all of this.
The bottom line is that these transmissions are perfectly capable of manual up- and down-shifting, if you want to or care to do that. However, in real life, it's MUCH MUCH easier and better just to put it in Drive and let the transmission do the work. Given that this is a very old design, four-speed transmission, it's really not set up for sporty driving, nor aggressive manual shifting a la a BMW M5 with a manual transmission, or even a Porsche with a Tiptronic.
The upside to the E500E transmission is that it is robust and durable and reliable. The downside is that it only has four gears, and is targeted more for regular and high-speed cruising rather than sporty driving, per the nature of the E500E itself.
Getting a bit more practical, per your question:
As said, let the car warm up a bit. Stock E500Es are set to start out in second gear, always.
If starting cold, it will hold the first shift (2-3) a bit longer -- higher revs -- to heat up the catalytic converters a bit faster. It will do this for the first couple or three minutes -- likely the first couple of stop signs you encounter after starting the car. Don't worry about this.
Shifting the transmission down and then into the "B" setting and back either up to 3 or 2 or D, at a stop sign/stop light, will move the transmission into first gear, where it will start from. Some folks modify their transmissions to always start in first gear, or have a switchable "first gear start" module that allows this. However, if you want first gear, it's just as easy in real life to activate the "B" switch and then move back up to Drive.
The max shift point speeds for each gear are indeed indicated by the dots on the speedometer, if you want to manually hold each gear to its max speed.
I would just experiment with the transmission, starting in second, first, and seeing what you like. Shift it up, shift it down going into corners, see how things work. Take care not to over-rev when downshifting. The transmission can easily handle manual downshifts.
I think you will find that it's much better and easier just to put it in D and leave it.
These transmissions have one weak point, and that is the reverse gear. This is all very well documented here on the forum. When the reverse gear setup is starting to go weak and on the road to failure, you will notice an out-sized delay in shifting into reverse. Normally a good transmission's reverse shift will be 0.5-1 second. When you get anything over 1 second, often times 2-3 seconds, you are very close to needing a rebuild. Eventually, the transmission reverse will fail outright. There are several good rebuilders nationally, also all documented here on the forum.
Typically this reverse issue will happen between 80,000 and 150,000 miles - generally toward the lower end of this mileage range on the E500E (80-100K). My own transmission has a 1.5-second reverse delay with 147K miles on the car. I have been nursing this delay for the past 10 years, but it is not yet to the point where it requires removal and rebuild.
These transmissions were spec'd originally with Dexron II transmission fluid, and do not like "modern" ATFs. Try to get a Dexron II or Dexron III compatible fluid if possible. You can go dino or synthetic with the ATF, but note as is typical you will likely get some modest leaks/weeping if you switch from dino to synthetic. Removing the transmission and doing a complete external re-seal will remedy this, and is not too difficult to do. It is a slight chore to remove the transmission yourself if you are good with a wrench, but is completely DIY-able if you ever need to do it (for example, to remove and ship it to a rebuilder).
