Commute between Birmingham Centre and Airport

I don't know how familiar you are with UK rail ticketing, but here are some details beyond what you've already been given (prices now updated for January 2017, unlikely to change throughout the year).

My source for this is BRFares, which is an unofficial site that displays data directly from the fares database. While a bit technical in operation (it displays all sorts of fares that aren't always available for purchase), it lets you see available fares from the perspective of the fares themselves rather than from the perspective of the journeys, which is convenient for this sort of question.

This is further complicated by the fact that the West Midlands has its own Passenger Transport Executive which set their own special fares. However, looking through these, I didn't find any that might be useful for you over and above what's listed on BRFares.

A couple of final things; it's commonly cheaper to buy a return than two singles in the UK, and this journey is no exception; but there's a caveat caused by the weird Network West Midlands fares. Also, there's no need to book tickets in advance, since for such a short journey there is no discount for doing so. You may as well buy them from the vending machine or ticket office at the station of origin each day.

I will assume, since you didn't mention it, that you don't have a railcard. Even if you are eligible for one, it won't be worth it for you to buy one unless you intend to make some more expensive domestic rail journeys in the UK in the same year.

(Note that no train gets overtaken between Birmingham International and Birmingham New Street so you should always take the one expected to depart next, even if it's an all-stations stopper!)

So, with that in mind, here are your options:

• Anytime Day Return with route Any Permitted: £6.90. Valid for a journey from New Street to International and back within one day (or up to 04:29am the following day), valid on any train running between these two stations (as well as breaking your journey at intermediate stations in the unlikely event you need to use them). Simplest option.
• Off-peak Day Return with route Any Permitted: £4.80. Same as above, but you cannot use this on weekdays at the busiest times of day (note that it's valid all the time at weekends). For this particular ticket, on the outward portion you can't leave Birmingham New Street before 09:30 or between 16:35 and 18:00, and on the return portion you can't use any trains timed to leave Birmingham New Street between 16:35 and 18:00 (which is a bit odd, as it implies you can travel during this time but only on trains that terminate at Birmingham New Street rather than travelling through...). I would go for this if I knew I would be safely avoiding travelling during those times, and certainly buy this for the weekends or you'll be wasting money.
• Off-peak and Anytime Day Returns with route Virgin Westcoast: I would avoid. They ARE slightly cheaper (£3.90 and £5.60 respectively) but make up for it by reducing your choice of trains from approximately nine per hour to approximately three per hour (those run by Virgin Trains only). The off-peak also has the same restrictions as above.
• Anytime Day Single: Like the return but only in one direction. Also comes in "Any Permitted" and "Virgin Westcoast" flavours, £3.70 or £3.10 respectively. Avoid unless it's the last journey you'll be making that day! There is no off-peak single for this route.
• Anytime Return: Not to be confused with Anytime Day Return. £7.20 or £6.10 for "Any Permitted" and "Virgin Westcoast" respectively. Valid for one journey out within five days of the date it's purchased for, followed by one journey back within one month of the date it's purchased for, with no restrictions on peak-time validity. Probably not useful for your journeys, but I'm including it just in case, and to highlight the possible confusion by the similar names!
• Evening Day Return (in some literature referred to as Super Off-Peak): £2.50. Valid only after 18:30, and again only for one day. A single is available for the same price. If you plan to leave before 09:30 but come back after 18:30, buying a combination of an Anytime Day Single and an Evening Day Single might be your cheapest bet (£6.20 vs £6.90 for the Anytime Day Return).
• Group-discounted tickets DO exist. There's a daytime group discount called GroupSave that gives you 1/3 off the anytime and off-peak day single and return fares for three or more people travelling together, BUT I believe London Midland is the only operator in the area who accept it, so once again it limits your choice of train (to about four per hour). Type GS3 into the discount box on BRFares to see the prices per person with this discount.
• There's also an evening group ticket that doesn't suffer from the same operator restrictions as it's a product of Network West Midlands. It's £6 for up to four people travelling together, and has the same validity as the evening singles and returns.

So, TL;DR: I would buy off-peak or anytime day returns as required (with route Any Permitted), each morning from the ticket machine, except for the case where you're leaving before 09:30 and returning after 18:30, in which case I'd use a combination of an Anytime Day Single and an Evening Day Single. I would forget group restrictions except where you'd normally use an evening single, when you might get a better price by buying an evening group return instead. If this is all too complex for you and money isn't that tight, just go for the Anytime Day Returns!