It's a broad question, but I'll take a stab trying to provide the essentials for a good start. Also see this post which contains important additional information:
Best telescope for the viewing of Nebulae, Stars and Planets
There are many, many factors involved in choosing a telescope. You seem settled on a dobsonian. That's not the only possible choice, but a dob is a good scope to start with.
You are correct that aperture (diameter of the primary lens or mirror) is the primary driver of telescope performance. Bigger aperture = better performance. However, performance is not the only thing you should worry about. Think of a car - do you only care about speed when buying a new car?
Any size scope will show you something. I've a 50 mm (2") achromat refractor that does double duty as finderscope. I can push it up to 100x magnification, and it will show lots of craters and mountains on the Moon, will show Jupiter as a disk with two equatorial belts, will show the rings of Saturn, will show the M13 globular cluster (barely) and the M31 Andromeda galaxy (again, barely). Not bad for a little scope.
A small 150 mm ... 250 mm (6" ... 10") dob will show you more. When seeing is good, you see several belts on Jupiter, and even individual curls on the big belts. Saturn's rings start to show the divisions, such as Cassini. These days Mars is at opposition - such a scope will show you the polar ice cap and the major land features. M13 fills the eyepiece with a dust of stars. The Orion Nebula looks big and clear even in the city. The Ring Nebula is visible, etc. Under a very dark sky, you can see (barely) the 3C 273 quasar, at 2.4 billion light years.
A mid-size dob (12" ... 18") will show you even more. More details on the planets, when seeing permits. More faint fuzzies (galaxies, nebulae, star clusters) especially out and far away from the city.
A large dob (20" and over) is an amazing instrument. On those very rare occasions when seeing cooperates, Jupiter at 1000x in such a scope is unforgettable. Also, under a very dark sky out in the desert, point it anywhere and the sky is full of faint fuzzies.
So, which size should you choose? It depends on price, maintenance, and ease of use (size, etc). Let's talk about maintenance.
As a dob owner, there are two things you must never forget: collimation, and cooling.
It means "keeping all the optics aligned and centered". The mirrors will move around a bit, and in time will lose collimation. As a dob owner, you have to collimate the scope once in a while (and every time before you observe high-resolution targets such as planets). Once you learn how to do it, it takes about 2 minutes, it's super-quick and easy.
This is why collimation is important:
This is an introduction to dobsonian collimation:
Here are some simpler collimation techniques, which you can use until you acquire all the tools:
Also see the manual that comes with the scope.
If you keep the scope in the house or garage, and then take it out for observation, the primary mirror will be much hotter than the night air. As a result, air convection will appear on the mirror and distort the image - just like the shimmering you see in the summer along the hot walls of a house under sunlight. The mirror is too "hot" for the ambient.
This is what happens:
To combat convection, take the scope outside at least 1 hour prior to observation, and let it cool off. Very small dobs (up to 6" or so) will probably work fine just with passive cooling. Small-medium and above (10" or above) basically require a fan on the primary mirror (forced cooling).
Always collimate and cool off the scope before you do an observation requiring high resolution (when observing planets, the Moon, double stars). If you don't need a lot of high resolution (like when observing galaxies, nebulae, star clusters) then collimation and cooling are not so important.
If you keep your dob collimated and cooled, it will provide great views of the planets. There is this myth out there that dobs "are not great for planets", and refractors are somehow magically "so much better for planets". This is not true. The myth exists only because of the large number of people out there who keep their dobs in terrible shape; those poor scopes underperform by a very large margin. Don't be one of those folks.
Let me just draw a line in the sand for you: An 8" f/6 dob is a good start overall. Why is that?
At 8", it is big enough that it will be seeing-limited most of the time. In other words, it is so big, atmospheric turbulence will limit the resolution more often than the size of the scope.
It is big enough to show you some faint fuzzies even from the city. All the Messier objects should be visible no matter where you are.
It is not too big to fit on the back seat of a sedan car, if you want to drive to a place with a darker sky. It is not too big to move around in the backyard easily.
The mirror is not so big and thick as to mandate active cooling (fan on the mirror); you can pretty much get by with passive cooling only - take it out at least 1 hour before observation. However, a fan may improve things a bit anyway (you need to experiment with that, but it's not a priority at this size).
At f/6 it's not too hard on the eyepieces, and it's not too hard to collimate. A shorter scope (f/5 and below) produces a steeper light cone; you need higher quality (and price) eyepieces to use it, and collimation requires much better precision.
Finally, the price is right - a few hundred dollars, which is affordable for quite a few people. Take care of it and it will last you a lifetime.
Now, if you can afford it and think you can deal with the steeper learning curve, could you buy a bigger instrument? Sure. 16", 18" - the wallet is the limit. Such big dobs are made differently - they are called "truss dobs". They don't have a big tube where everything is contained. Instead, the tube is replaced by a truss made of a bunch of poles - looks like the arm of a crane machine.
When you're done observing, you disassemble it. It all fits in a sedan car. It's not too heavy for transport, because you handle each part separately.
However, such an instrument is usually f/5 or less. Collimation is much more difficult. You need a coma corrector (like a Paracorr) to fix coma at the edge, and that's additional money. You need better eyepieces. Cooling doesn't work unless you use active cooling (fan). And it's just more expensive overall.
Can you start on a large dob like that? It's not impossible, but it requires more money and a much more significant commitment. This is not a toy that you can discard after two week-ends.