diplomacy question

About influence, it really depends on who you are, and what your goals are. The primary purpose of sphere's is to get access to another nations market (and their resources!). There are few, if any, countries which are actually autarkies in game. So having a sphere of countries with resources you need is highly advisable. You can let all others go, although you may want to keep them in case you need their resources later, want to prevent their resources from falling into another's hand, and give your country an export market. Furthermore, having friendly relations with a country allows you to intervene on their side in a war. A not to be underestimated ability.

You don't really need to look at the diplomacy screen all the time. You should have a diplomacy section in the bar on the right which will inform you of your influence and any competing influence.

About industrialisation. It really depends. Some countries are just naturally slow to industrialise, while others are really quick. Aside from that, some ruling parties allow you to build industry directly (mostly socialist and communist), but mostly it is a matter of letting the game build them for you. To do so, you'll need capitalists who have money (i.e. low taxation). There are other factors, but I'm not familiar with what they are.
So use your national focus to encourage capitalists if you have few, or use it to encourage industry in states which do have them.

Depends on what industry you have/wish to develop. If you go to the production screen, you can see what resources you yourself produce, as well as what you export and import.

in the Trade screen there are little tables on the bottom which show what you import and what you export. You can organize by money, quantity, or item. If you want to improve RGOS you import a lot of (or spend a lot of money importing) follow these steps:

1. Organize by money spent on imports
2. Go to RGO map mode abd click on a province with that RGO
3. Zoom out and look for provinces that are clustered together with that RGO or more green provinces with that RGO (preferably both)
4. Check Political mapmode; if Civilized nation, focus on sphereing, if uncivilized, conquer. (Sphered countries give only a certain portion of their RGOs, but civilized nations have tech bonuses and unruly populations. Uncivs don't have tech bonuses therefore when conquered you apply your own tech bonus.)

To prioritize factories, I find there are different approaches that can be made roughly corresponding with a supply side oriented approach or a demand side oriented approach.

Supply:

1. Check which RGOs you get the most money exporting from
2. Build the most profitable factory that uses that RGO as an import int the state that produces that RGO the most.

Demand:

1. Check which consumer good is in highest demand (most money spent importing)
2. Build a factory for that consumer good in your most populous state.

Supply approach maximises your profits from RGOs (useful for nations with a net export of in demand RGOs); the second approach maximises the happiness of your pops.