Yes Vicky 2 is not the easiest game to get into it. Hopefully I can help you a bit.
Skipping the video at start is indeed done by hitting the escape key. Though I think that if you rename the "movies" folder (in the main Victoria 2 folder) it should stop playing it.
I haven't played the tutorial, so I cannot comment on the it.
About influencing. If you go to the diplomacy screen you should see something akin to this:
(Not my image,
source)
In the bottom half of the screen you can see (from left to right) a list of countries, a 'prio' column, the flags of the Great Powers, a 'sphere of influence' colomn, various rankings (prestige, industry, military, overal rank), an opinion column and a relationship column. These allow you to sort the countries.
To influence a country you need to assing it a 'priority' by clicking on the three bars under the priority column. The more priority you gave (assign bars to) a country, the more diplomatic influence you get. Under the flags of the Great Powers you can see how much influence each Great Power has in that country, while the flag in the 'sphere of influence' column displays to who's sphere the country belongs. To add a country to your sphere, you first need to turn them 'friendly' (see the 'opinion' column), which you do by clicking the 'increase opinion' button (which spends 50 of your diplomatic influence in that country). Note that there are 5 levels of opinion; hostile, opposed, neutral, cordial, friendly (and 'in sphere'). So trying to get a country opposed to you into your sphere will be difficult. Once somebody is friendly to you, and you have 100 diplomatic influence, you can add them to your sphere by pressing the 'add to sphere' button (and in doing so spending 100 influence).
The amount of diplomatic influence you get in one country is influenced by a number of variables, hover over the priority bar to see how these variables stack up. Also note that the more countries you try to influence at once will decrease the amount of influence assigned to each individual countries.
Sometimes you will be in competition with another Great Power over a particular country (Egypt, Panama/Colombia and the German and Italian minors mostly). You can spend your influence points to sabotage their actions. The 'discredit' option will reduce the speed at which 1 other Great Power gains influence in that country, the 'expel advisors' option will remove all diplomatic influence of 1 particular GP in that country, the 'ban embassy' option will remove all influence of 1 GP and prevent them from gaining any for 1 year and the 'decrease opinion' option will reduce the opinion the country has of 1 GP (so for example it will no longer be friendly, but cordial). Lastly, if a country is in the sphere of another country, you can remove them from that sphere if their opinion of you is friendly and you have 100 diplomatic influence.
One final note, any GP can join in the defensive wars of a country it is friendly with as long as its wargoal is 'status quo' (or no war goal at all).
I think that is all. Am I missing anything?