(1) In game you will probably want to use an addon because the Blizzard API that tells you what quests you have done is extremely basic, and there is no Blizzard API that tells you what quests remain. Here are some links to addons that can help:
(a) The two addons Wholly and Grail are used together to provide the most accurate quest data:
(b) The kitchen sink of quest helpers, Carbonite, provides many more features than just the quest information:
(c ) For those interested in only quests associated with Loremaster:
(d) Another seemingly inspired by EveryQuest, Quest Completist:
(e) For those only interested in map pins for quest givers, QuestHubber:
(f) And the outdated, inspiring EveryQuest:
(2) If you are willing to use a web browser, you can get a raw feed of quest information from Blizzard's servers using a URL that is formed using your server name and your player name like the following:
Of course this is not very useful for most people, so other sites massage the data and present it to us nicely like:
However, even using this information from Blizzard will not provide the most accurate picture as there are quests that cannot be obtained because other quests have already been turned in. These links do not show that type of information since the Blizzard APIs do not return it. This is where external information about quests is required. If you have the quest ID that Blizzard uses, you can look up more quest information by using the proper quest ID in the URL that follows:
To get this quest ID it is usually easiest to use an addon that shows the quest ID in game.
And if you really want this wowhead quest information in game, there is an addon called LightHeaded that takes the web data and makes an addon out of it:
Hopefully this helps.