FWIW, I (mostly) love driving a car with a DSG transmission. Looking at this thread, some people are commenting on DSG transmissions in particular, and some about automatics in general. I have comments to make about both aspects.
There will always be some people who see driving an automatic as giving up something. But the truth is, we haven't had complete manual control of cars for a long time, so if that's what you want (why?), you're already out of luck. The engine has an EMU means you don't have to worry about a choke setting for a cold engine, and means that it isn't just your foot on the throttle that governs what the engine does; you have power steering providing (variable) assistance to your steering (which is itself nonlinear); ABS does cadence braking for you, and ESP controls the car's traction and stability in difficult situations; and, even on the most basic car you have turn signals that click off by themselves. An automatic transmission is just the continuation of these ideas; in most normal driving situations, the car can pick the right gear at the right moment all by itself. It knows far more precisely than you do where the optimal shift points are for power or fuel economy. And, the fact that the computers for the engine and transmission can talk to each other also allows for rev matching and other tricks that result in smoother gear changes.
Some people also think that if you drive a manual, that makes you focus more on what the car is doing, but I think that's wishful thinking. Regardless of transmission, a good driver knows how their car behaves, and understands both what the car wants from them and how to get the car to do what they want. And at the other end of the spectrum, a driver can be oblivious to their car, whether it's an automatic or a manual.
Almost all automatics, whether they are traditional (planetary gear) or DSGs do give you ways to control what the transmission does. They almost all let you force a downshift using the shifter, will downshift on a kickdown, and upshift if you left up on the gas pedal for a moment.
Myself, I learned to drive on a manual, drove a conventional automatic for the last eight years, and then this year I got a Golf TDI with a DSG transmission.
In general, I agree with people who say that the DSG combines some of the feel of an automatic with some of the feel of a manual. There is software trying to make choices about how to control the transmission, such as how much to engage the clutch(es) and when to change gears, etc.
Compared to a conventional automatic, the DSG changes gears mostly sequentially. If you're in 6th gear, you can't change down directly to 4th gear. The transmission can change from 6th to 3rd, but it is likely to have been expecting to change down to 5th and so setting up for a change to third isn't something it can do on a dime. The +/- controls on the paddle shifters or shifter lever also only let you go up or down one gear at a time, so it's not an option to change more than one gear at once. Manuals obviously don't have this issue, you can change into any gear you like, but the same is true for planetary gear automatics; if the transmission wants to shift from 6th to 4th, in principle it can.
Conventional automatics also have a torque converter. The torque converter allows hill-holding, slow forward creep at idle, and allows the car to have a lot of torque when pulling away from a dead stop by letting the engine roar into its power band immediately. The DSG has to manage creep by playing the clutch, is more likely to roll back on a gentle hill, and has to rely on the torque of the engine alone for pulling away from a dead stop. Some people say that torque converters lose power, but they all have lock-up clutches these days, some of which are pretty sophisticated.
But the DSG transmission does mostly work really well. Compared to the conventional automatic I used to drive, there was a learning curve though. Early on, the car often launched like a rocket when I wasn't really desiring that, because I was used to being heavy on the pedal. But that issue quickly vanished, so either I learned the car or the car learned about me (or both).
Mostly the DSG does a really good job of lining up the next gear and then seamlessly and smoothly chaining into that gear, bit that does mean that on the occasions where it isn't as fast or smooth you do notice. One situation I've noticed where my DSG struggles a little is in some cornering situations; I come into a sharp corner, brake, and then want to power out of it, it sometimes feels a bit wallowy and misses its mark, finally changing down too late to really be useful to me. I think what happens is that it wastes about a second dithering before it finds the right gear, but in that situation the second it takes matters. One option is not to power out of corners so hard; that way it won't need to change down. But the other way to deal with it is a general tip for the DSG:
Tip: If you know you'll want the transmission to change down but you don't want to assume full manual control, drop it into sport mode a little before the change. Thus, as you approach a sharp bend, change into S, power out of the bend, change back to D. You can flip between sport-mode and drive as often as you like; to me, S means “I'm going to need some serious power in just a moment or two”.
My other case for taking control is mostly a fuel economy thing; I'll change up to 6th at 30 to 35mph. So long as you're not going uphill, it won't lug the engine, and the transmission itself will decide to make the change eventually, but I'm just hurrying it along a bit.
Sometimes I use the paddle shifters to change down for engine braking, but not nearly so much as I did with my conventional automatic. The software for the DSG is smart enough to pay attention to your braking and engine brake automatically.
Tip: If you do use the paddle shifters on the steering wheel, you may want to give full control back to the transmission. One way to do that is by holding + for a couple of seconds, but that's not very useful if you just used - to change down for engine braking. For that, I use what I call “the shifter wiggle”, shifting it right into tiptronic mode (as if I was going to use its +/- controls rather than the ones on the steering wheel) and then immediately back left into regular D.
(Random DSG oddity I've noticed: Sometimes when I'm combining a sharp turn and accelerating from (near) zero, it only partially engages the clutch. The driving is smooth and the power feels right for the accelerator position, but the revs are totally out of whack from what I'd have expected—much higher. First time it ever did that it freaked me out a bit, but I recognize it as a thing it does. I'm sure it has its reasons.)
But mostly it's smooth, powerful, smart, and does the right thing, and I like it.