There are various sites that use various algorithms to estimate the required fee to get your transaction included in a block. I would suggest https://bitcoinfees.earn.com/ that is both simple and gives you enough information to estimate how large a fee to include depending on how quickly you want your transaction to confirm. /edit - you should review the chart to decide how much to pay for your transaction fee

To answer your other question, yes you can include a very small fee and it might take a long to for your transaction to confirm if it ever does. After several days, most nodes will ‘forget’ your transaction if it doesn’t confirm but it will remain valid as long as all the inputs remain unspent. This means someone may broadcast your transaction long into the future if it is still valid. This means you should keep your private keys, including backups if you ever even try unsuccessfully to send coin between wallets.