You need a casus belli that reduced the infamy cost. For example, imperialism (later on in the game), or holy war (earlier in the game) incur only 50% infamy for annexed provinces.
I was hoping for CBs much earlier though, like <10 years... Read that PU seems to be the best low BB way to expand, could I get PU by marrying into Firenze and Ancona (and possibly Siena too) and then claim throne and inherit? Probably got to be very lucky to pull it off without huge penalties. Was thinking of releasing the non-core provinces (Pisa and Siena) as vassals then annex them back when Milan becomes emperor so I won't get the prestige hit and Liberation CB against me, but Pisa and Siena both have universities and I don't want to give up the extra research and income. A bit weary about attacking Papal states too, will I be excommunicated if I take Roma by force?
You can get good advisors by increasing your cultural tradition and then hiring the advisor of choice. You can do so either by building specific buildings (academy of fine arts gives +1/year) or certain decisions (like introduce heir/national epic, etc). You can also simply pump it up temporally by using magistrates, either by commissioning paintings (free) or endorsing culture if you have above 50 prestige. The latter costs 10 prestige but gives +8 cultural tradition per magistrate. Once you bring it up nice and high and hire an advisor, there's a good chance you'll get a very skilled on. That's your best bet for a method early on in the game, I think.
Didn't know you can stack them up using multiple magistrates, I will try that then, thanks.
The diplomatic map mode shows non-cored provinces in stripes, you can use that and click on the country of choice. For economics, there is an economic map mode that uses red (low) -> green (high) color schemes to indicate province income (relative to your other provinces).
Was hoping for more, including map mode for countries other than my own. Oh well, can't have everything I guess. Couldn't find the economic map mode, the closest seems to be the revolt map mode with the green and red zones.
The emperor asks as a random event when you have non-core provinces. Either get a core or suck it up and deny it. The event only comes once per province. Unfortunately there isn't really another way, unless you want to leave the HRE and remove the provinces from the Empire.
So it's only once per province, that's good to know because I was worried that I will get -5 prestige and -2 stability hit every year. Is it likely for the emperor to use his liberation CBs? Northern Italy looks defendible but I am still worried about fighting with big shot like Bohemia.