If you roll the money from a 401k which I'm assuming is tax deferred, to a Roth, which is not, you will have to pay the taxes. But then the money will grow tax deferred and tax free. Since tax rates are the lowest they've been in decades, you might reconsider this strategy, as Roth's are a much better place to grow your retirement money than any other qualified account.
If your new employer is matching in a 401k, I would advise to invest only up to the match, and then put the balance of the money you can in your Roth and pay the taxes now. If you're a successful investor you will pay far less in overall taxes doing it this way.
Once you leave your company, they must let you roll your 401k monies out of their plan, and to avoid taxes and penalties they must go into a tax deferred IRA type of plan. To avoid penalties, they must go into a Roth type of plan. You can consolidate multiple accounts into one account, provided they are not being administered by your company and subject to their rules.
If you need more into, please email me.