jmm2112 has it right. Pick a company like vanguard or fidelity and roll it into an IRA. The reason to pick this method and NOT roll it into your new employer 401K is that you will be changing jobs again in the future. It is good to have one place to put all your money---including IRA's Roths and working cash. In addition, vanguard and fidelity offer a wide range of investments so that your IRA is working for you throughout your working career.
If you can afford it you should be funding ROTH's with after tax money as well. Keep these in your umbrella account with vanguard or fidelity as well.
Just call the company of your choice and they handle to whole rollover.
I suggest you talk with someone with Vanguard, Fidelity or one of the other large financial companies. I believe there are some advantages to having an IRA rather than a 401K. Depending on your age a Roth IRA might be even better.
if your new job has a 401k plan, roll it to the new 401k. otherwise, the easiest thing to do is contact the 401k administrator and have then convert your 401k to an IRS and keep all the funds the same if possible
Yes. Do a direct rollover. Pick the IRA custodian of your choice and they will guide you.