Is it possible to make a transaction with no transaction fee?

As Raghav Sood noted in a comment, transactions with no transaction fee are not transmitted by standard Bitcoin nodes - so your zero-fee transaction normally won't reach a miner. In theory you could make some special arrangement with a miner but it's hard to see why they would want to do that.

Is it possible to pay more fee to prioritize your transaction?

Choosing a higher initial fee tends to result in quicker confirmation. If your wallet supports Replace By Fee (RBF) you can use that subsequently to improve the likelihood of a quicker confirmation.