How best to back-up data on Big Sur to be transferred to new computer?

Don't buy the model with a 256GB SSD, get the next one up with the 512GB SSD otherwise your migration will be a mess! If you ignore this advice believe me once you are done, you will see why I recommended that you do it!!!!. Once you have the 512GB model on order simply do the following for a successful migration. By the way, current Apple Refurbished Macs come with Mac OS Sonoma, not Big Sur.

When you buy a new Mac the easiest, quickest and most reliable method to migrate from the old Mac to the new Mac is to simply connect the old Mac's Time Machine External Hard Disk to the new Mac using a USB cable before you turn on the new Mac. You will likely need a USB A to USB C adapter, you can find them on Amazon for $10 or less. When you start the new Mac for the first time it will startup Setup Assistant (SA), SA will get to a point where it asks if you are migrating from another computer, select Yes  and follow the on-screen prompts and it's done! It is THAT simple.

If you have already done SA on the new computer then simply connect the Time Machine External Hard Disk using a USB cable and launch Migration Assistant (MA) which is located in Applications - Utilities - MA. Then follow the on-screen prompts. 

The ONLY difference between MA and SA is MA will create a new user account that you need to log into.

Both SA and MA migrate applications (not 32 bit apps), data files & settings. They do not migrate Mac OS itself. Some professional apps such as Adobe and Microsoft apps may require re-installing their apps as they will not allow them to be migrated. As long as you kept the security keys for installing their apps, it is not an issue. 

Finally, never EVER use Wi-Fi when doing Setup Assistant or Migration Assistant. Why, it simply fails in most cases. Please read and follow the directions in Setup Assistant and Migration Assistant. If you do not have a Time Machine backup of the older Mac then create one! If you are not aware of Time Machine, please read Back up your Mac with Time Machine and finally if you need an External Hard Disk to use as a Time Machine External Hard Disk then many experience users on these forums prefer the OWC Mercury Elite Pro due to it's reasonable price, high level of quality and ease of use. 

1. Is Time Machine the easiest way to transfer all of my stuff from the old to the new computer?

Yes.

... If I leave all of my music on the external drive the rest will fit on the smaller 256GB hard drive on the new computer.

Ok, if you say so. In my opinion 256 GB is a paltry amount though.

When considering a new Mac, you really ought to consider your present storage needs and then double it. You have had your existing iMac for about eight years, and it is reasonable to expect at least as much useful life from its replacement. How much additional storage do you think you will need over that period of time?

It's easy for me to spend your money though.

2. If so, do I need to "partition" the external SSD in order to keep my music files separated from the Time Machine back-ups?

I strongly recommend that you do not partition the drive. What happens when that TM device fails? You're much better off purchasing two or three separate, redundant, "slow", inexpensive, disposable hard disk drives for Time Machine than you are purchasing one large SSD and partitioning it.

Speed is utterly irrelevant for Time Machine. When the TM backup device fails, throw it out. Buy another at your convenience, since you will be left with a minimum of one that still works.

3. When I get the new iMac I am assuming I will want to upgrade the OS from Big Sur to Ventura or Sonoma...will the (assumed) Time Machine back up work with that?

Yes.

#4 If I save my music to the external drive now, when I use the Migration Assistant can I tell it not to migrate the music files?

Yes. You can choose to exclude Music, and simply copy that entire folder to the external drive some other time.

You simply need to save up for a bit longer. Also, why aren't you taking advantage of Apple's 12 month same as cash offer?????????????????

https://www.apple.com/shop/browse/financing?afid=p238%7CsG7bknzHB-dc_mtid_1870765e38482_pcrid_674076641966_pgrid_68125343220_pntwk_g_pchan__pexid__&cid=aos-us-kwgo-iphone--slid---product-

Buy what you need, don't try and force too much data into too small a space. It's short sighted and frankly just foolish!