Q: How do I upgrade my iPhone 4s to iPhone SE?

The sim card can be taken care of at the carrier's store or the Apple store when you upgrade.

As for getting your content on the new device I would suggest using an iCloud backup. You can sign in to iCloud then in settings > iCloud you can choose to sync your data and then make a backup.

When you get the new device, in the setup assistant you will be able to sign in to iCloud  and setup the device from your iCloud backup.

Hope this helps!

Good luck!

Apple has a tip for moving from one iOS device to another: Transfer content from an iPhone, iPad, or iPod touch to a new device - Apple Support

Just call the carrier and they should send you a replacement SIM. You cannot use a blank SIM, because there is no such thing. A SIM has a phone number programmed into it. If you get a new SIM you will also have a new phone number.

No such thing as a blank sim.  The sim card contains cellular carrier information for your phone number, and carrier settings.

Your carrier can easily replace your current sim for one that will fit an iPhone SE.  This also almost always done for free.

If you cannot for whatever reason go to your carrier's store or online help form them to send you a sim, you can attempt to cut down your sim to size so it fits in an SE.  However that should be a last course of action.

There should be no issues with your carrier for changing sim card size.

You can call your carrier and ask them for a new SIM.  They should be able to ship it to you pre-provisioned for your account, or you can call them and have them provision it over the air once you have it.

Transferring your number and account to a new device and SIM over the air should take any carrier no more than a couple of minutes.

Google "cutting your own sim card to a nano sim card" and you can have your phone ready in no time