1st time playing - Not sure to understand the goal of the game.

To get Influence, set a priority on any nation you want to influence. It's the little box with three lights on it on the Diplomacy window, where all the countries are listed.

Crises are being resolved in white peace so often because the great power backing the nation that started the crisis looked at the forces arrayed against them and decided that the small prestige hit of giving in was worth not having to lose a war.

Hover over the option to form an alliance so you can see what the problem is. Is your infamy too high? Are you too weak? Do they already have allies?

Anyway, as Italy, shouldn't you be making sure you have a good navy? Fortify your borders with France and Austria, and certainly make sure they're defended, but you need to look to Africa to expand. At 1870 you've probably missed your chance to scramble for some colonies of your own (since I'm not sure if you have the fleet or the techs you need), so you'll have to resign yourself to picking off other peoples' colonies. Go for states that produce resources that your factories need; that will make you more money, and keep you on a more secure footing militarily and economically.

Who has the Netherlands allied with? You should totally take their Asian holdings from them once you get a big enough navy to crush them at sea. Take the Suez Canal first if someone else has it.

What are your National Focuses ("NFs") set for? You may want to set them for soldiers in your most populous regions if you want a larger army.

You can most likely take a colony or two from Netherlands, and if Spain joins them, take something from Spain. It is possible, though highly unlikely, the two nations will be able to coordinate a substantial attack against you. Most of the Netherlands' troops will be in the E. Indies, and Spain's troops will likely be split between Cuba, the Philippines, and European Spain. If you can't outmatch Spain's and the Netherlands's navies, try and get military access through France.

To answer the title of your post, your goal can be whatever you want it to be. Do you want to have the greatest prestige, industry, military, or any combination of these? Do you want to create a communist utopia (or dystopia xD)? Do you wish to conquer all of North Africa? Do you want to re-create the borders of the Roman Empire? How ambitious you want to be is up to you.

It's tough to give war advice without seeing specifically what situation you're in, but if you could get a navy large enough to take care of the Netherlands's navy, you might be fine. Spain has to come across France, through mountainous terrain, to hit you. The Netherlands either has to come through the NGF and Austria, or through the Strait of Gibraltar. That's an easy choke point for your navy to defend.

There are only eight great powers. Find the one who doesn't have a dance partner, and start stepping on people smaller than you.

You need fleet and naval bases to get colonial points, but it may be too late to build it in 1870.

Building your own Sphere of Influence is also very important, you get access to rare resources and military alliance.

There are 8 GP, what are they in your case? It should be GB, USA, France, Russia, Prussia/NGF/Germany, Austria, Italy and one more, who is it? If I correctly understood, Spain, Ottoman Empire or Netherlands. You can influence two others and include them into your SoI, then they will take your side.

China and Japan usually became GP after westernization (you can check their actual score in ledger), so some countries may loose their status. Will it be you? If you will remain GP, you can make then an alliance with a country, which partner fell out of G8. If you will become secondary power, it can be even better, since you will be able to make alliance with anyone you want and start crises against Austria.

There is a tech or invention (I don't remember exactly), which will allow to ally more then one GP, but later, in 1890s, IIRC.

The goal of the game is to just have fun playing a semi-historical game whilst you get immersed in the atmosphere and period. Otherwise its pretty much anything else you want it to be, you set your own goals in this game. For many people it will either be getting westernised, getting as high up in the country rankings as possible, getting to be a great power, or just even surviving till the end year, or all of of these.

These are all usually the initial goals for me until someone takes some of my land off me, then it becomes a personal vendetta against my sworn enemy.

I'm playing my 4th game, so far I've "lost" my first 3 games. By "lost" I mean I didn't reach the goal I set for myself:
- First game, as Brazil, wanted to be a Great Power. Failed, only got to like 12th.
- Second game, as Columbia, wanted to be a Great Power and form the Great Columbia. Failed, got to like 18th.
- Third game, as Qing, wanted to be a Great Power and kick Russia's arse. Failed (this I think because I was using the POD mod, which I read later makes playing China almost impossible), rebelions kicked my arse and undid everything I had done in the first 20 or 30 years.

So there you see, you set your own goals, just don't be too optimistic in the beginning
P.D. Playing my 4th game as Prussia, and after playing the other 3 countries, this feels like playing in god-mode. My goal for this game was to form Germany and be #1 GP. Having issues with forming Germany as it's hard to see what provinces I need, and countries I've sphered with my provinces then became GPs.