I’m not going to touch immigration, work permits etc, because it varies greatly - I’m assuming you figure it out. For skilled workers with work experience there usually is a fairly painless way to get all you need.

Continuing to work:

• your employed has to have presence in the country you are moving to, or
• they have to handle your employment through an intermediary, like deel.com, or
• you have to transition to independent contractor (potentially legally dicey if you are a contractor in name only)
• if your company doesn’t support fully async work don’t move more than 8 time zones away - that way you’ll still be able to join some meetings

Moving is the simplest part:

• Lightweight & cheap option: pack a backpack/suitcase like you were going on long vacation. Buy plane tickets. Rent Airbnb at the location for a week and use that week to rent a place to live. This option is similar in cost to moving to a different city within a country with extra costs being $2000-3000 for travel and initial week at destination.
• Everything and kitchen sink is not much more expensive: 10k gets everything you own professionally packed, stuffed in a 20 feet shipping container, shipping across the ocean, customs and delivery and unloading (but not unpacking from boxes). 20 feet container is enough to take everything in a large, packed 2 bedroom condo including furniture.

At destination you will need:

• work permit / work visa
• local equivalent of social security / tax number / sometimes both - file a form, sometimes pay a small fee
• a business (if you are going independent contractor route)
• bank account

Vast majority of the info you need will often be available on the embassy website of your destination country.

Source: over the 20 years of my career I moved across the ocean twice with my family and worked from a total of 4 countries.