Diffie-Hellman computing public key with a large private key Given is Generator $g = 5$, Multiplicative group $F_{10007}^*$, Private key = 7347.
Then public key is $5^{7347} \bmod 10007$ right?
On a calculator this gives an overflow error, how am I supposed to calculate the public key when a private key is a large number?
On a calculator this gives overflow error, how am I supposed to calculate the public key when a private key is a large number?

A standard calculator cannot handle that, and we don't expect it. If you have a programmable one you can use square-and-multiply as in levgeni's answer. This, however, will fail when the calculator cannot handle integers (the modulus) larger than the calculator's word size.
There are better and easy approaches

Use WolframAlpha 5^7347 mod 10007
Use SageMathCell 5^7347 % 10007
Use Python's pow that takes a third parameters as modulus pow(5,7347,10007) and uses modular-square-and-multiply. Online