There's two ways of doing multi-player connections, Peer-to-Peer 
and Client-Server:

  With Peer-to-Peer (each player sends messages out to all the other 
players), you'll have to figure out how you'll do collision detection
- the best way to do it is for the collision detection for each player's 
car to be done by the player's program, as only one machine can make a 
decision for one object. For things like power-ups, maybe it would be best
for the Server to handle those decisions (and effectively controlling all
objects other than player's cars, and messaging is done Client-Server).
Also these messages would have to be sent reliably, as you couldn't afford
to lose information about picking up a power-up etc. 
  To prevent graphical problems with higher lags, you could add an 
artificial lag to your player's movements, say based on the lag to the Server
minus, say 50-100 ms. 
 
  With Client-Server, (you send all messages to the Server which then sends 
them all out to the players). This is good because your player doesn't move
until it receives the message back from the Server, so all objects move
around the same on all machines (assuming lags don't vary too much).
(With a racing game this is less of a problem). Collision detection is
easy because the server does it all, but if the server is an Amiga,
then that would put extra load on the machine (as it would have to run
the game as well)- you would need a powerful machine for it. Also you
would need a decent modem because the server has a lot of data coming
in and out- it takes a load off the individual player's machines
though.
   The main problem is, that Quake servers etc are usually permanent at
an ISP and have a direct link to the net, so they don't add any
appreciable lag to the messaging; but if the server is home computer
with an analogue modem (which in your game is likely), then the message
going through the analogue modem and out again will add a large amount
to lag (for a 300ms lag, half of that will be the modem! So if your
game was Client-Server it would add another 150ms to the lag!)