I Paid $600 to a pet transportation company and after some talks in which due to some wrong information we were asked to pay an additional $300 i asked to cancel and was told that they would withhold 50%. A contract was Written but not signed by either party prior to payment is there any way i can get my money back?
If there was a written contract, the fact that it wasn't signed is not relevant. While a signature is evidence of agreement with the terms there are other ways that acceptance can be indicated: like you paying them $600.
Where the error is fundamental to the performance - e.g. you needed shipment to Alaska and they were offering shipment to Alabama, the contract would be void ab initio. That is, it never happened and everyone needs to be returned to their original positions as far as possible.
However, in general, an error by one or the other party in their understanding of what was agreed does not invalidate the contract. For example, if you told them it was a "small" dog because it was small for a Great Dane but under an objective classification, it is, in fact, a "large" dog the contract must be completed and either you or they wear the additional cost of doing so.
Whether they are entitled to ask for additional payment "due to some wrong information" depends on who took the risk under the contract for its correctness? Barring a specific term, the risk usually lies with the party that provided the "wrong information" but some contract will assign the risk for one party's errors to the other party - subject to a requirement to act in good faith.
If they are not entitled to additional payment, they have to perform the contract for the original fee. If they are entitled, then you have to pay a reasonable price increase - you are not generally entitled to cancel.
All of this turns on the specific terms of the contract and the exact nature of the "wrong information".
Consumer Protection Law
CPL in your state or their state or both will almost certainly have something to say about this beyond common law rules of contract.