After completing a level in Story Mode, then Free Play Mode is unlocked, like the other versions of the game. In Free Play Mode any character that has been unlocked can be taken into the level, with the intention of using the various Characters and Abilities to find all 10 Minikit Canisters in every stage.
The first suggestion as always is to complete Story Mode for all three episodes so that a Jedi and Shortie Character can be unlocked. After this, purchasing Vader (Sith) and a TIE Fighter and the Probe Droid for Vehicle Stages. Finally, like the GBA Version, Characters are unlocked by completing the Minikit in every stage, unlocking more abilities. A suggested Order for Free Play Stages is as follows, so you don't have to replay stages:
The Blockade Runner
Rescue the Princess
Mos Eisley Spaceport
The Asteroid Field
The Trench Run
Hoth Battle
Betrayal Over Bespin
Death Star Escape
Escape From Echo Base
Cloud City Trap
Jabba's Palace
The Great Sarlaac Pit
The Speeder Chase
Battle on Endor
Heart of the Death Star
Destiny
The list of Character Abilities in LEGO Star Wars II for DS is as follows. Note that not every Character Type is technically useful, but is listed to complete the list:
Astromech - Astromech Droids get a bit more use than Protocol Droids. They can access Astromech Panels and Hover around as well. Eg. R2-D2, R5-D4.
Blaster - Blaster Characters use a gun for combat, attacking from a distance. They can use Melee attacks up close and can use Grapple points when applicable. Useful characters, and the most abundant type in the game. Eg. Leia, Han, IV Luke, etc.
Bounty Hunter - Blaster characters that get one extra ability, the ability to use Bounty Hunter Panels. Eg. Dengar, Greedo, etc.
Boxer - Boxer characters primarily use their fists to attack, with no other options available to them. Otherwise, they're largely uninteresting. Eg. Leia Jabba's Prisoner.
Droid - A Droid. Doesn't do much otherwise but hey it's there! Eg. Gonk Droid.
Explosive - Bounty Hunters that have the ability to throw Thermal Detonators to destroy harder objects. Eg. Leia Bounty Hunter, 4-LOM, etc.
High-Jump - Characters who have the ability to jump higher than even the Jedi, giving them more height than anyone else. In this game, combined slightly with Hover, as the Fetts are the only ones who have this type.
Hover - These Characters have a short hover ability instead of jumping that allows them to cross larger gaps. Eg. The Astromech Droids, The Fetts.
Jedi - Bringers and peace and justice in the galaxy... or at least, they were. All Jedi have the ability to Double Jump, use the Force, and attack with a Lightsaber. Useful characters for general use. Eg. V-VI Luke, Obi-Wan, Yoda, etc.
Imperial - The Imperial Army has an extra use to them. They can use Imperial Panels to get into places other characters can't. Eg. Stormtroopers, Vader, etc.
Physical - Characters who can attack with weapons but don't fit into any other proper category. Eg. Gamorrean Guard, Biths.
Protocol - Protocol Droids use their advanced intellect to use Protocol Panels. Besides that, they're very much Droids. Eg. C-3PO, K-3PO.
Shortie - Characters that are shorter than others, giving them access to passages otherwise unusable, generally Boxers and Practical's otherwise. Eg. MSE-6, Wicket, etc.
Sith - Very similar to Jedi, but gain an extra few abilities depending on the character, they can all use Sith Force, while Force Lightning and Force Chokes depend on the character. Eg. Darth Vader, The Emperor.
Walker - Basically useless. Good for one thing. Walking. That's it. Eg. Han in Carbonite.