You must copy the spells from the found book into yours so you can use them
Per the Multiclass spellcasting rules:
You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class.
You will still need to copy the spells from that spellbook into yours in order to be able to use them. Please see this question for more detail, but you'll need to spend the time and gold to move them into your spellbook in order to use them
That means that for the Wizard spells, you'll be a Level 1 Wizard. That gives you 3 Wizard cantrips and 6 spells in your newly gotten class spellbook.
At 1st level, you have a spellbook containing six 1st-level wizard spells of your choice. Your spellbook is the repository of the wizard spells you know, except your cantrips, which are fixed in your mind.
You'll be able to prepare 2 of the wizard spells in your spellbook (Level 1 wizard plus an INT modifier of +1.) These will be in addition to the spells you know as a Sorcerer.
Copying limitations - 1st level spells only
The rule around copying a spellbook still apply here (emphasis mine):
When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a spell level you can prepare and if you can spare the time to decipher and copy it.
Copying that spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation.
For each level of the spell, the process takes 2 hours and costs 50 gp. The cost represents material components you expend as you experiment with the spell to master it, as well as the fine inks you need to record it. Once you have spent this time and money, you can prepare the spell just like your other spells.
Because of the limitation that you can only copy a level of a spell you can prepare, you will only be able to copy level 1 spells from that book. But you can keep that book on hand if you continue to increase your wizard levels.
Now, as as a Sorcerer 6/Wizard 1, you'll have the following spell slots from multiclassing:
- 4 1st level spell slots
- 3 2nd level spell slots
- 3 3rd level spell slots
- 1 4th level spell slot
This effectively lets you upcast your prepared 1st level wizard spells, but you still can only prepare 2 first level wizard spells.
You can cast either Wizard or Sorcerer spells with this (but use your ability modifier per class.) You'll also need a requisite component pouch or focus:
Each spell you know and prepare is associated with one of your classes, and you use the spellcasting ability of that class when you cast the spell. Similarly, a spellcasting focus, such as a holy symbol, can be used only for the spells from the class associated with that focus.
Just a reminder that if you have any ritual spells in your spellbook, you can cast those as rituals without preparing them.