Once his installment was settled, the pope set about making many changes.
Alexander was a reformer. He despised the corruption prevalent in some ranks of clergy, and so he proclaimed, “We are well aware that morals have notably fallen back. No longer can we tolerate the way in which the former salutary measures instituted by our predecessors to keep sensuality and avarice within bounds have been violated so that we fall headlong into corruption. …Even though we were only in a lesser position as cardinal under our predecessors, Pius II, Paul II, Sixtus IV and Innocent VIII, we always wanted to see the present licen[s]e of morals restrained by new constitutions. That is why at the beginning of our Pontificate we wished to give this matter priority over all others.” 
Alexander soon nominated a commission of six cardinals known for their piety and had his “Bull of Reformation” prepared. The Bull outlined several proposed disciplines: new limits on the sale and transfer of Church property, the restriction of cardinals to one bishopric each, cuts to their households and property, bans on their participation in popular secular events, limits to their funeral expenses, and a firm zero-tolerance policy for clergy caught accepting bribes or utilizing concubines. 
Michael de la Bedoyere says of this, “We need to bear in mind in assessing the true character of Alexander Borgia this extraordinary capacity to see so clearly the right path and to will it with all his heart…” 
Unfortunately, he was never able to push through those reforms. This was the biggest tragedy of Alexander’s reign – and it is one that surely upset him to a great extent. There was always something in the way: battles with the French, political squabbles, clerical revolts, and so on.
Nonetheless, he quickly accomplished much.
He was able to cut the rampant crime in Rome by implementing measures like the appointment of new prison inspectors and commissaries. He also held public audiences where anyone, no matter their status, could lodge complaints and receive justice from the Vicar himself, further ingratiating the Church to Her people.  In addition, he planned a new constitution to outline the rights and liberties of his citizens. The prisons became better-managed and the new judicial districts, each run by accountable magistrates, brought about increased safety and effectiveness. 
Alexander also immediately set himself apart by his erudition. This was nothing new, but he now had a wider audience. He was “so familiar with Holy Writ, that his speeches were fairly sparkling with well-chosen texts of the Sacred Books.” 
The Italians, as a whole, liked him. But seeds of their republican spirit had begun to spread, notably among the aristocrats.
The Sforzas were certainly a thorn in Alexander’s side. Ascanio was greedy. Ludovico was crazy, putting whole territories at risk to further his ambitions. Caterina, “the widow of Count Girolamo”, on top of her other crimes, even tried to poison him!  This family was quite dysfunctional, despite its occasional good work in Milan.
The Orsini were treacherous and especially powerful. Virginio and the rest of his family continually plotted against the pope and Cesare. Alexander wisely kept Cardinal Giambattista Orsini as a hostage while the French were about, to prevent further betrayals at that critical time – and that worked, for a while. The cardinal was arrested again, in 1503, and held along with Franciotto (the protonotary apostolic) and other members. Their properties were seized for a time in that year.  The Orsini have also long been suspected of assassinating Alexander’s brother, Don Pedro Luis, after he fled to Civitavecchia for safety, when his uncle was near death. 
As for the Colonna, weaker at the moment but influential still, Prospero (a military man) had to be imprisoned. At one point, in June 1501, their properties were finally taken, in the name of the Holy Father.  Had the Orsini and Colonna not been feuding incessantly, they might have done irrevocable damage to the Papacy.
And then there was Naples, ruled by Ferrante I (aka Ferdinand I). Ferrante had many enemies, because of Naples’ importance to others, and he was ruthless in response to any perceived slight. After he realized that Alexander could not secure his position, he fiercely plotted against the pontiff.  He died in 1494, “without the light of grace, without the cross, and without God,” still an enemy of the Church. 
The Medici, alternatively, were not bad. Giovanni, a cardinal and the future Pope Leo X, at just 16, is rumored to have loudly denounced Alexander’s election. Such a statement is unlikely, however, in light of both his youth and his family’s cordial relations with the Borgias. Cesare was on particularly good terms with the Medici – “the greatest confidence existed between him” and Giovanni’s older brother, Piero. 
Lastly, there was Savonarola. This fiery preacher condemned his age and he was filled with conviction against perceived wickedness, though there was nothing particularly special about him. Even his admirers recognize that, while he had holy intent, “there are no grounds whatever for believing that Savonarola’s prophecies and revelations are to be attributed to any special divine inspiration.”  While he had his quibbles with Alexander, it must be noted that he accepted a plenary indulgence granted to him at his death by the pope, subtly assenting to his authority.  Importantly, this friar set the stage for future discussions on papal infallibility. 
Alexander’s enemies were definitely better at one thing: churning out propaganda. “Literature happened then to be a rising power; it was mostly under the influence of the enemies of the Holy See (because princes were rich and [p]ontiffs were poor) and the venal writers of Florence or Venice”.  Unfortunately, he did not feel the need to correct these rumors, laughing them off and saying, “Plenty of things are said of me, but I take no notice.”  “It never occurred to him to attempt to curtail the liberty of speech or writing in Rome.” 
The pope had to put up with a lot. “Perhaps no [p]ontiff had ever been exposed to such a combination of wicked and abandoned princes; or ever one defended himself so nobly and successfully.” 
1. de la Bedoyere, p. 25
2. John Julius Norwich, Absolute Monarchs (ISBN 978-1-4000-6715-2), p. 268-269
3. de la Bedoyere, p. 27
4. Burchard, p. 54
5. de la Bedoyere, p. 90
6. de Roo, Vol. 2, p. 273-274
7. Burchard, p. 114-115
8. Burchard, p. 169-173
9. Ludwig von Pastor (edited by Frederick Ignatius Antrobus), The History of the Popes, from the Close of the Middle Ages, Vol. 2 (John Hodges, 1891), p. 476-478
10. Burchard, p. 147-148
12. Hibbert, p. 55
13. Gregorovius, p. 48-49
14. de la Bedoyere, p. 81
15. Michael Davies, 1996 (http://store.keepthefaith.org/orthodoxy-heresy-and-reform-part-3-girolamo-savonarola)
16. Gregorovius, p. 128 (Quote: “In 1497 Hieronymus, then in Cesena, composed a dialogue on Savonarola and his ‘heresy concerning the power of the Pope.’ The kernel of the whole thing was the fundamental doctrine of the infallibilists; namely, that only those who blindly obey the Pope are good Christians.”)
17. The Dublin Review, Vol. XLV, September-December 1858, p. 342
18. von Pastor, Vol. 6 (Kegan Paul, Trench, Trübner, & Co., 1898), p. 113
19. von Pastor, Vol. 6, p. 115
20. The Dublin Review, p. 343