With a Compose key:

• 
  Compose , " → „
  


• 
  Compose < " → “
  


• 
  Compose > " → ”
  


• 
  Compose < ' → ‘
  


• 
  Compose > ' → ’
  

You can enter the last two characters in either order, at least in most locales. There may be locales where it only works in the order Compose < " works or only in the order Compose " <.

If you need to press Shift or AltGr to enter some of these characters, you'll need to hold those modifiers down as usual. You can press and release Compose before the subsequent characters, or you can press Compose and keep it down, as you prefer.

Most keyboards don't have a key marked “Compose”, but you can configure one and your environment may already have one configured. On a PC, a popular choice is the “Menu” key between the right Windows key and the right Ctrl key. The keyboard settings of most desktop environment include an option to make this key or some other key a Compose key.

Some ‘smart’ [1] quotes without using Compose [2] (and without remembering Unicode codepoints):

AltGr + 9 = ‘

AltGr + 0 = ’

AltGr + [ = «

AltGr + ] = »

AltGr + Shift + [ = “

AltGr + Shift + ] = ”

These work on any desktop, provided you have ISO_Level3_Shift assigned to your AltGr key and are using a keyboard mapping like the US International Keyboard with AltGr.

For Compose combinations, check Gilles' answer (or the XCompose file for your locale). They, in turn, are dependent on your locale and/or the presence/settings of your ~/.XCompose file.

[1] ‘Typographical’ is probably more appropriate. ‘Smart’ refers to the ability of a word processor to automatically use the correct character in a quote pair when you type the ASCII quotes ' or ".

[2] You can, of course, have both Compose and AltGr enabled. I do, and it works a treat.

I redefined my keyboard layout for good and I simply press alt-key + ; or ' to get: “ ”. Works in every desktop env.

There are many choices how to do it -- for example, you can use character map app (present in Gnome and KDE for sure) to get any character you want.

For anyone that runs across this: the Compose key for curly quotes didn't work for me until recently. I'd tried it in 2-3 major distros with GNOME 2 and changing the assigned key didn't help, but when I tried with KDE 4 in SimplyMepis and set it to use CapsLock it started working just fine. So trying it again or switching environments, distros, or keys might be worthwhile.

Linux Ubuntu users can type smart quotes with ...

AltGr+v / AltGr+b ( “ / ” )

... or ...

AltGr+Shift+v / AltGr+Shift+b ( ‘ / ’ )

On any keyboard layout that I know of, producing these quote marks
requires using the third level of some keys. This is typically
accessed by holding down the AltGr key as a shift key.

(I’m going to focus on typing these symbols directly, as opposed to by
using the compose key or using the “NumPad entry” method, since these
symbols are typed so often that they deserve a way to be accessed
directly.)

Many European keyboards seem to use AltGr+V (“)
and AltGr+B (”)

As for US keyboard, the standard US keyboard does not have these
symbols. It only has two levels (unshifted and shifted) for
alphanumeric keys, and none of them contain these symbols. However,
there is the “US international” variant, on which these symbols can be
accessed on Shift+AltGr+[ (“) and
Shift+AltGr+] (”). As you’ll see if
you try to type this, this is impressively inconvenient, as you have to
hold Shift with either hand, AltGr with the right
hand, and then either [ or ] with the right hand
(†1).

US International is named us(intl) if you use a tool like Setxkbmap:

setxkbmap "us(intl)"

†1: The French quote marks («») gets the more convenient place one
level down (AltGr+[ and
AltGr+]), which is curious since I’d think that
the English curly quotes would get a more convenient placement than
them.

More convenient typing on US International

There is an option misc:typo (“typo” as in “typographical”) which can
be combined with us(intl) since it only redefines some level 3 and 4
keys. In particular, it maps “ and ” to
AltGr+K and AltGr+L,
respectively. This seems much more convenient than the keys that
us(intl) uses. To use this option with us(intl):

setxkbmap -option "misc:typo" "us(intl)"

Another thing that can be done is to define some alternative or
additional level 3 modifiers, so that the double quotes can be typed
more comfortably. According to man
xkeyboard-config,
Xkb provides quite a few ready-made options:

Option: Description

lv3:switch: Right Ctrl

lv3:menu_switch: Menu

lv3:win_switch: Any Win

lv3:lwin_switch: Left Win

lv3:rwin_switch: Right Win

lv3:alt_switch: Any Alt

lv3:lalt_switch: Left Alt

lv3:ralt_switch: Right Alt

lv3:ralt_switch_multikey: Right Alt; Shift+Right Alt as Compose

lv3:ralt_alt: Right Alt never chooses 3rd level

lv3:enter_switch: Enter on keypad

lv3:caps_switch: Caps Lock

lv3:bksl_switch: Backslash

lv3:lsgt_switch: <Less/Greater>

lv3:caps_switch_latch: Caps Lock; acts as onetime lock when pressed together with another 3rd-level chooser

lv3:bksl_switch_latch: Backslash; acts as onetime lock when pressed together with another 3rd level chooser

lv3:lsgt_switch_latch: <Less/Greater>; acts as onetime lock when pressed together with another 3rd level chooser

Personally I think that CapsLock makes for good level 3 switch:

setxkbmap -option "lv3:caps_switch" "us(intl)"

If you prefer the misc:typo keybindings, you can of course include
that option as well:

setxkbmap -option "lv3:caps_switch" -option "misc:typo" "us(intl)"

You can use alt codes Alt+0147 for “ and Alt+0148 for ”. Hold Alt and type the numbers that follow.