Does calculating how many bitcoins I have take longer as time goes on? I have read that the blockchain is a series of transactions that record one person giving bitcoin to another.  Please correct me if this assumption is wrong as it is the basis for my question.  So, over time, doesn't it take an increasingly long time to figure out if you can accept Bitcoin from someone because you will have to go through every transaction in the history of the blockchain to validate that the person giving you Bitcoin does indeed have that Bitcoin in the first place?
With a naive implementation, yes, but in practice, no, it is not a problem.

This is because any developer that knows what they are doing is smart enough to create a database which, at the very least, records each transaction (or its position on disk) and its verification status. So when you receive a new transaction, instead of having to walk back through history, you just need to look back to the referenced previous transaction. As part of validating a transaction, you check if the previous transaction was valid. So if a transaction is marked as valid in your database, then you can be sure that the transactions that preceded it  are also valid. Thus there is no need to actually look through history and re-validate those transactions.

Since it ultimately just comes down to a simple database lookup, it does not become harder to validate new transactions.