First of all sorry for a bit late reply... summer is a busy travel time for us
Regarding your question, there are two (and a half) possibilities to consider: a) purchase in advance on
TrenItalia website discounted 'MINI' offer tickets for legs Venice-Rome and Rome-Naples with high-speed trains, if your date of travel is not very close, then there are good chances to get a good deal (MINI tickets start from only 9 euro); b) use slower regional trains. Here you won't need to buy any tickets in advance, but just do it before departure already in Italy. Obviously traveling by slower regional trains will take longer (especially in case of Venice-Rome), but much cheaper than full fare tickets for high-speed trains (kind of the same option, but slightly different ticket combination is just to visit Belgrade train station and buy there international SCIC-NRT ticket Gorizia-Venice-Rome-Napoli, it should cost 52 euro and will be valid for one month with unlimited stopovers, but valid again only for regional trains, as result you just substitute domestic Italian tickets for regional trains with one international through ticket)
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