After being named dictator by the interrex L. Flaccus late in 82 BCE, Sulla reinforced the number of senators, which had dwindled sharply from it's traditional 300, and increased that number to 600 by the inclusion of many equestrian-rank persons, especially those of the Italian elite (newly minted as Roman citizens following the recent Social War). In order to maintain this number, quaestors would now automatically qualify to enter the senatorial rolls, and the quaestorship was now a prereq for the praetorship and consulship. This ensured a steady supply of new senators to replace ones who died, were killed in battle, or were murdered in political strife. In order to bolster the ability of the senate to administer a growing empire, especially in the East, Sulla increased the number of yearly quaestors to 20 and the number of praetors to 8.

These were all moves designed to shift political authority back to the senate as a body, but not necessarily to the traditional members of the senate (the old Roman elite). The inclusion of "Italian" (now rural Roman) elite was an important step in the development of a pan-Italian "Roman" elite identity in the 1st century BCE. For this aspect, see especially Santangelo, Sulla, the Elites, and the Empire (Brill 2007).

You mentioned the tribunes. Part of the dictator's restoration of senatorial authority meant a severe curb on the abilities of the tribunes of the plebs. They were no longer permitted to introduce legislation, and they were disallowed from running for higher office. This meant that ambitious young elites could no longer use the tribunate as a chance to make a lot of noise and gain political notoriety before moving on to higher office (like praetor and consul). These two measures both neutered the People's legislative power and took away their voice, and both were doomed to eventual failure. Sulla also prohibited the tribunes from exercising a veto in senatorial meetings, which was downright contrary to the mos maiorum and the ancient compromises set in place at the beginning of the Republic. Sulla hated tribunes and had been obstructed/annoyed by them on several occasions.

The final "major" reform was Sulla's transfer of the courts back into senatorial control. The battle over who would fill the juror's seats during trials had been going on for quite a while, and would continue after this. Since the Gracchi, the courts had been stuffed with equestrian jurors; they were now returned to those of senatorial rank. The back and forth of the influence over the courts requires an entire answer all on its own. For here, just note that Sulla here went against the equestrians in favor of the senate.

By none of these acts did Sulla seek to retain power. In fact, only a year later, he voluntarily gave up his autocracy and "retired" to his villa at Puteoli. Sulla considered all of his acts justified, even the most bloody and capricious, and most of his career as a Roman statesman involved his persistent notion of being wronged, passed over, or otherwise not appreciated. When he, the rightfully elected consul, was preparing to depart Italia for a war against the invader Mithridates, his command was stripped from him by the antics of Marian tribunes. Though technically legal, the situation was a bit absurd, and his adoring soldiers agreed. They were happy to march with him on Rome to rectify this silliness. He then left for the East as planned and prosecuted a successful war, punished the enemies of Rome, made significant additions to the Empire, and brought back a criminal amount of loot to Italy. Upon landing back in Italy, he found that not only were his accomplishments not recognized, but that again the pesky Marian tribunes had turned the Republic upside down. He was a Cornelius, a member of an ancient clan, and the war and bloodshed that followed must have seemed, to him, like an unfortunate necessity to "right the ship." He probably thought he owed it to himself, but also to the memory of his ancestors.

The bulk of Sulla's reforms did not survive him for very long. Already in 75 BCE, most of the tribunate's shackles had been lifted. It was Pompey and Crassus, Sulla's old proteges, who finished the dismantling during their co-consulship in 70 BCE. Sulla's hope to end the political circus by his reforms had not panned out, and the 70s were marked with strife just like the 80s and the 90s and 100s had been marked. Seeking compromise, Pompey and Crassus again changed the courts: now there would be 1/3 jurors of senatorial rank, 1/3 of equestrian, and 1/3 of tribuni aerarii, which were like equestrians but of lower status (and very much able to be influenced by both senators and equestrians). This proved feasible.

The irrefutable modern authority on Sulla and the aftermath of Sulla's policies is still Gruen, The Last Generation of the Roman Republic (Berkeley 1974).