Tests... well the official answer :
According to the official website (I discovered it a few minutes ago...) the ticket for collective tests in Monza is 10€ (3€ for under 18, free for under 11). It allows to go everywhere but it the pitlane/paddock, you need a particular pass for that but I honestly don’t know how to have it, I never asked.
BTW, that’s only for collective F1 tests, if there’s only 1 team (or even 2) there’s no ticket to pay it’s just like any regular day (you have to pay only if you want to enter with your car from the principal gate, but why should you ?).
At the official website of the Autodromo you can find the list of events of the month with all the indications you need :
http://www.monzanet.it
Then the unofficial answer... as far as I remember I never paid for a collective test... and not because I jump the fences, simply there are a couple of gates that they close only for the GP, you just have to know where they are and pass from there. I have the suspect that they leave them open on purpose, it has always worked this way. From there you can go everywhere but in the grandstand in front of the pits because there’s a ticket control there.
Monstrobolaxa : what your dad tells to you is what my dad tells to me since I was born, and they are right, F1 is a tv sport, actually that’s true for most of motorsport races, if they’re available in tv then stay at home you will see a better show. The only race I believe are better at the track are endurance races.
About the “good thing”, one of my preferred experiences is to stay in one of the metallic “bridges” crossing the straight line between Ascari and Parabolica. You see nothing because it’s a closed metallic structure, but when the cars pass under you, you can really feel them with your whole body as the structure vibrates. There are a few further spots where you can really feel similar good things, the tunnel under the main straight line, the “tunnel” right before Ascari etc.
Anyway the feeling is the same on test day and race day so...