You can do multiple callouts, as long as all of the callouts are done BEFORE any of the DML operations.  I know - that probably doesn't help you because you are most likely looking to perform a callout using the result of a DML operation.  Might I suggest in that case, looking at calling @future methods. The Operation could be framed as the following:
1. Perform callout #1 to obtain data.  
2. Insert callout results into a Salesforce record.
3. Call '@future' Method and pass the 'Id' of the record you just created.
4. in your '@future' annotated method, perform your second callout and the rest of your operations as needed.

'@future' methods can perform callouts as long as the method is documented with the '@future(callout=true)' annotation.  Please be warned, however, that this process only works for 2 callouts. Unfortunately, because you CANNOT call another @future method when you are executing inside of one, you cannot use this process to perform another callout after a DML once in your first @future method. Also, please note that technically calling '@future' methods spawns another transaction context, so in-memory variables and such will not be available to use in your '@future' method unless you can re-create them.
see https://developer.salesforce.com/docs/atlas.en-us.apexcode.meta/apexcode/apex_classes_annotation_future.htm (https://developer.salesforce.com/docs/atlas.en-us.apexcode.meta/apexcode/apex_classes_annotation_future.htm" target="_blank) for more information.