What is the best, DBMS-independent way of generating an ID number that will be used immediately in an INSERT statement, keeping the IDs roughly in sequence?
DBMS independent? That's a problem. The two most common methods are auto incrementing columns, and sequences, and most DBMSes do one or the other but not both. So the database independent way is to have another table with one column with one value that you lock, select, update, and unlock.
Usually I say "to hell with DBMS independence" and do it with sequences in PostgreSQL, or autoincrement columns in MySQL. For my purposes, supporting both is better than trying to find out one way that works everywhere.