Disclaimer: I'm not from Italy but from Germany - I've been living for a while in Italy in the past but for tax purposes was (fortunately! - as being German I know much better how to deal with the German tax authorities and their forms) still classified as German, so my insight into the Italian banking system is limited.
Firstly, I'd distinguish:
- whether the broker is domestic or foreign, and
- whether they offer trading at domestic and/or foreing exchanges
For a beginner, my recommendation is regardless of whether you are legally allowed to use foreign online traders to go with a domestic one. It doesn't hurt to look around whether you can find a suitable one that also offers trading at foreign exchanges.
The reason for this is that (at least in Germany, but I expect the same in Italy) the broker will already deduct due taxes and will give you formal declarations which may even tell you where in your tax declaration you enter those numbers. This very much reduces the risk of making stupid mistakes with your tax declaration and also lowers the work you need to put in to fulfil your legal requirements.
Before moving on to having also a foreign broker, there's the intermediate step of owning foreign shares/bonds/... and finding out which foreign paid taxes you can deduct from your domestic declaration (tax treaty is the magic word) and how to do this.
Trading on a foreign exchange via your domestic broker really isn't that much different from choosing at which domestic exchange your order is to be placed.
Once you have experience with all this, you can find out about foreign brokers. From my Germany-based experience, it is allowed in general (and I very much expect that this is the same for Italy, due to EU regulations) but there are certain regulations to be fulfilled.
Here, even the necessary declarations for sending money to foreign accounts are done automatically by broker/bank as long as both sending and receiving account have the same owner (they notify you that they send the required notification to customs).
I'd not be surprised to find more burocracy about this in Italy, though. (Also from the recent experience of an Italian customer coming along with financial transparancy declarations paperwork in order to pay my bill)
The important thing to keep in mind is that you'll probably have to make sure yourself all business with the foreign broker is declared and taxed appropriately. Not sure how much work this is in Italy, in Germany as long as you stick to simple stocks or bonds and the foreign country also has similar taxation concepts that yield some kind of tax document this is not terribly difficult.
One of the things I learned in Italy is that even within EU tax laws can be totally and unexpectedly different. I'm talking of concepts like stamp duty and the tax on checking accounts which we don't have here. So if the taxation concepts are different, it's going to be something between difficult, a lot of hassle, or downright impossible to deduct the foreign taxes from your domestic due.
And of course, unexpected foreign rules for taxation make it more difficult to calculate how much will be due and check whether automatic deductions are at least plausible (e.g. in Germany, for the tax calculation on capital gains there are rules what losses may be balanced against which gains and which may not be balanced against each other - I don't expect they are obvious for foreigners)
This boils down to (unless you are of the monetary weight to employ dedicated tax lawyers) foreign brokers coming at a certain risk of unpleasant (= costly) surprises. I mentally treat this just like any other fee, but what I don't like is if this is unpredictable.
Fortunately, there are online brokers who are specialized on foreign customers and do have experience with this and also how to best transfer money between your domestic bank and them. They can also answer your questions and give sensible predictions on the amount of paperwork needed.
I definitively recommend that checking how much experience they have with foreign/Italian customers as one of your evaluation criteria.