Generally for a card-not present transaction (i.e. MOTO transactions) you will need cc#, expiry and possibly the CVV (aka CVC2 etc). You can obtain the first 2 from a card-swipe as this in the track data. CVV is printed on the card.
Name on card doesn't matter so much. Unless your acquirer and the cardholder are using address verification, but you can find that between ^^, it may have white space padding which you can remove.
The part you want is track2 NNNNNNNNNNNNNNNN=1210 where NNNNN=card number PAN, and 1210 = Expiry date.
Even if track1 is empty (which sometimes it is as it's not used in processing), you will still get the ;?, so you could use the index of the second ; as start of the string and = as the end of the cc# string. With the 4 characters after the = as the expiry.
I would advise getting the card holder to sign something in record of the transaction otherwise they could dispute the card and do a charge-back.
And not all credit cards have exactly two tracks, some uses three tracks.
Only track2 is used for processing and has a standardized format.
Debit cards can't generally be processed (unless they have a visa-debit card or something).
P.S. you shouldn't store cc data in plain text, so try and keep everything in mem or strong encryption.