They do a pretty good job of introducing the world and story in ME2, if you have not played ME1. That being said, I feel that picking up ME1 for $20 is well worth it and will definitely enhance your ME2 experience.
For example. certain characters will not appear in ME2 if you did not play ME1. Make sure you get ME1 and ME2 on the same platform too. You can't transfer saves from different platforms.
IMO, you definitely want to import your Mass Effect 1 character into Mass Effect 2. All of the major choices that you made in Mass Effect 1 will be reflected.
For example, at one point you visit the council on the Citadel, and that plays out completely differently depending on whether or not the original council is still alive (same with the choice of who you nominated for the council, Anderson or Udina). The same goes for who you killed on Virmire (Ashley or Kaiden) and if Wrex is still alive.
The same is true for the side quests, to a lesser extent. However most of these only have the follow up in the form of an e-mail to your private terminal on the Normandy, but it's still cool to know that your choices had a lasting effect on the story.
If you had a love interest, it doesn't have a huge impact on Mass Effect 2. When you meet them, you share a brief kiss, but that's about it. You are free to have a new (different) love interest in Mass Effect 2, but apparently that will have some sort of impact in Mass Effect 3.
Bioware did an excellent job continuing the story you started in the first game. If you could make a choice in Mass Effect 1, it is reflected in Mass Effect 2.
As for importing characters, I'm not completely sure about this but I believe that you can import any play through for a given character (it's been a while since I imported my Mass Effect 1 characters). If you don't mind doing another Mass Effect 1 play through, I'd suggest starting again with your level 60 character (or whichever one you want to import), and make all of the decisions that you want to carry over to Mass Effect 2. You could always put the difficulty on easy to help things go faster.
If you want to know how a given quest affects Mass Effect 2, you can always check the Mass Effect Wiki.
If you don't import a Mass Effect 1 character, the game basically makes the big Mass Effect 1 choices for you. Also, some dialog that is present for an imported character is not present for a new character (mostly minor things, but some text wouldn't make sense if you didn't play the first game).
Finally, at the start of Mass Effect 2 you are able to change your class and physical appearance (don't worry, it makes sense in the context of the story). You can't change your sex or name, however.
Best Answer
When you start a new game in Mass Effect 2 without importing a save file from Mass Effect 1, a variety of default options are chosen to choices that could be made in Mass Effect 1. There is no way to change these choices in Mass Effect 2 however, it is possible to create a Mass Effect 2 save and then use an external editor to change the default choices.
In order to do this you will need to do the following:
Start a new game and create a new character (don't import)
Play through the introduction level and the initial cutscenes until you "wake up" and are able to save your game. Save and exit the game
Download and open Gibbed's Mass Effect 2 Save Editor
Open the save file that you just created
Under the "Player" tab within the "Plot" tab - tick the "Played Mass Effect 1" option
Go to the "Plot" tab to change any of the other options you want
Ensure that "End Game State" under the "Plot" section on the "Raw" tab is set to an option that is not "NotFinished" (ie: "LivedToFightAgain" or "OutInABlazeOfGlory")
Save the edited file ensuring you stick with Mass Effect 2's naming convention (Autosave.pcsav, Quicksave.pcsav, Save_000#.pcsav, etc)
Open Mass Effect 2, start a new game and select Import Mass Effect 2 Character, choose the file you saved and play...