Well I must admit swamp wasn't my first game with the mouse, I also used it in rail racer and in some experimental arcade games written in 2008 by a chap called Bryan. actually sinse those included audio tutorials and served as something as an interactive how to on mouse control it's a shame they're no longer around (I checked the page where they used to be and got a 404), one even involved being a cleric throwing spells at nasty monsters which was cool!   
Unfortunately, I can't reccord an audio tutorial sinse both swamp and Daytona need headphones to play and I can't reccord and use headphones at the same time sinse I use sterrio mikes (and playing swamp on speakers would be a nightmare).

I will however try to give as best text description as possible sinse I do know there are blind people out there who have trouble trying something new.

holding the mouse:

I always myself hold the mouse with my right hand (particularly because in swamp all the important keyboard controls can be activated left handed). I have my thumb along the left edge and the heal of my hand towards the back so that my index finger is on the left mouse button and my middle finger is on the right button. I can then slide the mouse along with my thumb and ring fingers. If I needed the mouse wheel I'd probably shift my index finger along a little.

Moving the mouse:

Because I am not the best spacially coordinated person, I prefer myself to use the edge of my desk to orient me (particularly because I have a good sized computer table). So when I'm playing swamp I actually have the back of my hand over the edge of the table which gives me a good streight edge to move along. I can then pretend essentially that the mouse is like one of those big lever rudder controls that some boats have that are like single levers which move in a groove left and right, this also makes it easy to pick the mouse off the table and shift it right if I need more left than my amount of room allows (though with the default sensativity in Swamp it seems okay). Sinse I have the back of the table as a guide to a streight edge, moving the mouse streight up and down is easy, which will do for things like operating the throttle in Rail racer where you need to push the mouse forward or pull it back.

For something like Daytona, I pick up the mouse and move it forward into a blank space on my desk where I have room to swing it around a little and do the circles and such. This is a little less precise, but that is why Daytona is a harder game, because your not just going in streight lines, your going in curves and circles, ---- but hay, making gold isn't easy .

I don't really think there is anymore I can say, sinse other than that it's all practice, practice practice! Maybe people should go and try a space invaders game that uses the mouse like Judgement day, or maybe the new shades of doom 2.0 on training mode (where the monsters stand still), to get the knack (sinse quite apart from the mouse Swamp is not exactly an easy game to begin with).

Hth.