How do I not get incinerated or cut down on the very first battle?

The sign that protects you is Quen, it absorbs part of the damage and prevents you from being incinerated. It is extremely useful, just know that your vigor doesn't regenerate while it's active.

In general, for group battles you should make liberal use of everything you have available. If you have time to prepare, drink a potion (e.g. swallow) and set traps. Use Axii to make one of the enemies fight on your side, distracting the other. Yrden sets up a trap that hurts and immobilizes an enemy for a short while, also very useful. And before you engage in sword combat put on Quen to protect yourself.

But the most important thing is to never get yourself surrounded! Try to seperate the enemies and use hit&run tactics to attack one enemy and quickly evade (double tap direction key or space) once the other enemies arrrive.

When the dragon is attacking, you need to move quickly. Once he's lit each shelter on fire, you'll take damage periodically. When you move to the next shelter, you have a brief space until he lights it on fire.

When fighting groups, make liberal use of the Block command (E). After a block, immediately counter-attack with a quick strike (left mouse button). Aard can also be useful as it will knock back a foe and interrupt their attack. Don't let foes get behind you if you can help it. They deal extra damage when hitting you in the back.

Also, to see all the tutorial tips, you can go to the Tutorial section of the Journal (default J).