The job description for the first flight of the day basically says this:
Chips
Charts
Checklist

Meaning:
Chips: Do the catering.
Charts: Set up the EFBs, fill the daybook and get out appropriate sectionals.
Checklist: Do the pre-flight.

Before EACH FLIGHT (including that first one, and thereafter):
Get the ATIS and clearance.
Program the flight route into the GPS.
Put the code in the transponder.
Do weight and balance sheet (except for luggage and fuel).

All of these should be done before the captain is ready to go; really about a half hour before the flight. Note that the time listed on the trip sheet should be (though often is not, as we know) wheels up time, not engine start time.

While the captain is going through start-up routines, the F.O. should:
Put the baggage and fuel lines on the weight and balance sheet.
Push the buttons to get the GPS on the correct pages.
Push the buttons to get the MFD going.
Set/check the radios and transponder.

During flight:
Put receipts on expense reports.
Put away used daybook charts.
Program future routes in GPS.

After each flight:
Clean aircraft (vacuum?).
Refresh catering.

End of the day:
Dump out coffee and ice.
Vacuum.
Tag the aircraft.

As a general rule, constantly be thinking about what things need to be done at any given time. Ask, "What else can I do?" and do it. It doesn't look good to be reading or wandering around the lounge if the clearance isn't picked up, or to be doing all of the weight and balance during the flight (or, say, three flights later: which would be illegal if anyone ever did that, but I'm sure nobody at Alpha does).

So, here's a checklist:

FIRST APPEARANCE
Chips (catering)
Charts (EFBs, daybook, sectionals)
Checklist (pre-flight)

WELL BEFORE EACH FLIGHT
Clearance
GPS: flight route
Transponder set
Weight and balance done (except for luggage/fuel)

DURING START-UP
Weight and balance complete
GPS up
MFD up
Radios to proper frequencies
Check transponder code

DURING FLIGHT
Weight and balance luggage line
Expense reports
Put away used daybook charts
Program day's routes in GPS

AFTER EACH FLIGHT
Clean aircraft
Refresh catering

END OF DAY
Dump coffee and ice
Tag aircraft