Your guesses are probably as good as any native speaker's. These expressions draw only a little bit upon English idioms; to understand their meaning mostly requires knowledge of Petrarch and his situation, along with some common knowledge regarding medieval Europe.
I'm just learning Latin, though, so I did a little searching to see if I could find enough context to determine the meaning precisely. Be warned that I'm only a beginner with Latin, and this is the first time I've ever tried to read Petrarch, so there are likely to be mistakes here, but here's what I found.
The passage is from a letter, dated May 30, 1342, to Johannes Columna, a friend of Petrarch's and a cardinal in the Catholic Church. Petrarch is advising Columna not to complain so much about his troubles. He spends pages trying to inspire Columna with stories of the sufferings of famous people from antiquity and how they found equanimity despite old age, poverty, and gout. Near the end of the letter, Petrarch describes at length his modest living situation in France, saying "If you come visit me, here's what you'll see." Here is the full sentence your passage is taken from. I quote it in Latin so you can see how much fun Petrarch is having with the sound and grammar of the language. I'm pretty sure he made up some of these words:
Videbis a mane ad vesperam solivagum, herbivagum, montivagum, fontivagum, silvicolam, ruricolam, hominum vestigia fugientem, avia sectantem, amantem umbras, gaudantem antris roscidis pratisque virentibus, execrantem curas curiæ, tumultus urbium vitantem, abstinentem liminibus superborum; vulgi studia ridentem, a lætitia mœstitiaque pari spatio distantem; totis diebus ac noctibus ociosum, gloriantem musarum consortio, cantibus volucrum et lympharum murmure, paucis servis sed multis comitatum libris; et nunc domi esse, nunc ire, nunc subsistere, nunc querula in ripa, nunc tenero in gramine lassatum caput et fessa membra proiicere; et (quae non ultima solatii pars est) neminem accedere nisi perraro, qui vel millesimam vaticinari possit suarum particulam curarum.
Yes, that's one sentence. Here's my attempt at a literal translation:
You'll see from morning 'til evening a solitary wanderer, a grass-wanderer, a mountain-wanderer, a spring-wanderer, a woodsman, a rustic—fleeing the footprints of humans, attending to daisies, loving the shadows, delighting in dewy caves and green meadows, cursing the cares of the curia, shunning the bustle of the cities, keeping off the thresholds of the haughty, laughing at the preoccupations of the crowd, standing at equal distance from joy and sorrow, idle all the days and nights, glorying in communion with the Muses, in the songs of birds and the murmur of waters—accompanied by few servants but many books—now at home, now going, now pausing, now on the bank of the warbling river, now laying his wearied head and tired limbs on the soft grass; and (which is not the least part of the relief) approaching no one except rarely—who could foretell even a thousandth part of his troubles?
A curia, as I understand it, is like the board of directors of a church. They meet and make decisions about finances, hiring, upkeep of the building, etc. Petrarch lived here, not far from Avignon, which at that time was where popes held court. So, the nearby curia was probably very busy, filled with politics and social climbers. Saying "cursing" is a playfully naughty way to describe disdain for those things, since they're at the very center of the Catholic Church. Here's an article about the modern version of that curia (which is now in Rome).
A threshold is the horizontal strip of wood in the floor beneath a door, which you step across when you enter. So, in context, avoiding crossing the thresholds of the mighty or haughty means avoiding visiting the important and powerful people of the city. As I understand the Latin word superbus, it connotes that they're arrogant and unpleasant. "The mighty" doesn't necessarily suggest that in English, but here, just after expressing disdain for life in Avignon, it might. Compare "Oh, how the mighty have fallen!" "Cross the thresholds of the mighty" suggests to me either invading their space (entering uninvited) or obeying them (entering because you were told to). "Crossing a threshold" in English is usually meant metaphorically, to describe an important transition (explained a little more here). I don't think the translator intended that, but the phrase suggests it. "Thresholds" here should be taken completely literally. I understand the Latin abstinere here to mean "keep off" (or "hold back from", "abstain from") their thresholds, which to me suggests not even asking permission to enter, because you're not visiting them at all.
In other words, the two phrases you asked about paint the busy life of a cardinal in the worst possible light. Petrarch is saying by implication, "Come visit me and get away from all that!"
By the way, it's not completely clear to me that Petrarch was describing himself. As I read it, the third-person description might be a way of saying "This is how your days would be while visiting me." The previous sentence is "You'll see what you desire: good health, [etc.]" "See" in Latin, as in English, can mean "achieve this result".