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 employer 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, shipped across the ocean, moved through customs, delivered to your new address and unloaded (but not unpacked 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.