Like Rich said, it's best to let a professional gunsmith handle the barrel swap. Mainly because the barrel will come as a blank piece of steel, other than a hole in the middle with rifling in it, and pre-contoured to the specific contour you ordered. It will need to be threaded, chambered, fitted, and have the headspace trued. While you have the old barrel off going through that process, you might as well have the smith true up the action while he's at it.
Yes, I would bed the action and fully free-float the barrel. Floating and bedding are 2 totally different things. Bedding keeps the action planted so it doesn't move around in the stock causing accuracy issues and POI (Point Of Impact) issues. Floating the barrel frees the barrel from any outside sources (the stock) affecting the harmonics it creates when a bullet is pushed through it at rapid speeds. If there's nothing external acting on the barrel to affect harmonics you will have much better chance of getting a truly accurate load developed.
So, whoever told you bedding was a necessity and floating is not, doesn't need to be giving rifle advice. And I wouldn't listen to much of anything else they said if someone made a comment like that to, or in front of me. Just saying. My appologies if it's someone you know really well, but my time is more valuable to me, than to listen to hear-say B.S.