How could Italy have a more succesful post-unification experience? In terms of military endeavours, territorial expansion and economic potence.
Simple- make sure Francesco Crispi isn't Prime Minister. Crispi's first term in office from 1887-1891 inadvertantly crippled Italy thanks to his obsessive Francophobia. By secretly joining the Triple Alliance, he ensured that the French would be furious when the news inevitably leaked out- worse, it did so during the negotiations over a renewal of the Franco-Italian trade treaty, which he subsequently walked away from. Crispi's subsequent escalation of the crisis into a full-scale trade war, which lasted a decade, was catastrophic, and essentially destroyed all realistic hope Italy had of being a 'proper' Great Power.
In 1886, almost half of Italy's exports went to France, and the French were also the single largest importer. This trade relationship was destroyed practically overnight, and did not recover until after WW2- in 1913, the equivalent export figure is only 9%. The value of trade between France and Italy fell from 444m lire in 1887 to 165m lire in 1888. Put simply, this had the effect of completely buggering the Italian economy.
This not only helped spur Italian emigration to the US (and for that matter, France), but it gutted the armed forces. In the 1880s, the Italians had a very modern and well-trained navy that was the equal of the French Mediterranean force if not slightly more powerful; by 1893 sailors were going without pay and the fleet was effectively left to rust. Lack of funds also contributed to the Adowa debacle.
Thanks to Crispi's demagogic, brutally ‘frank’ speeches, and his annual habit of orchstrating war scares, it was virtually impossible for the French Government to make concessions while he remained in power, even though he made some half-hearted efforts in that direction; he clung to office for just long enough to ensure the damage to Italy was permanent.
So, how do we get rid of Crispi? This is actually pretty easy; in 1879, his career was almost terminally damaged by the revelation that he was a bigamist. His comeback in 1887 was rather improbable and can be derailed quite easily; for the minimum number of butterflies, let's say that Giovanni Nicotera has his miraculous comeback three years earlier than OTL and supplants Crispi.
What then? Well, Italy would conclude agreements with Britain as OTL and probably flirt with the Triple Alliance, although not become a full member. Relations with France would be tense but far improved over OTL, and a renewed Franco-Italian trade treaty would probably be signed at some point or other, although it may take several torturous years of negotiation. Even if the deal falls through Italy would still be far better off ITTL.
In the long term, Abyssinia probably goes Italian, and will be heavily developed; doubtless some of OTL's emigration across the Atlantic will be diverted to the Ethiopian highlands. It's just possible that everyone gets ahead of themselves and an Italian column sets off into Equatoria to fight the Mahdi, although it's more likely that the desire to remain on good terms with the British will trump this. A bit of clever diplomacy might leave the Sudanese border rather closer to the Nile than in OTL however.
After that, we'll probably see enthusiastic interference in the Balkans and Libya ahead of schedule- maybe an Italo-Ottoman war c.1905 or so followed by the Balkan League piling on in a reversal of what we saw in 1911. Maybe a Chinese concession or two could be claimed in the wake of the Boxer rising too. In power bloc terms, a middle course between Germany and France would probably be steered, irritation at Austria gradually winning out over cordiality to Germany- perhaps an official Anglo-Italo-Japanese bloc emerges?
Either way, by the 1900s Italy is far stronger, more militarily powerful, claims a larger empire, has a larger economy, and is more self-confident. On the downside, removing the influence of Crispi may well introduce a fair bit of political instability beyond what occured OTL, but at least it removes a man who in a number of ways was the prototype Duce.