Not great to be honest, though if you're a craftsman, artist, or intellectual, you may well be spared. In sieges like that of Baghdad, certain groups were invited to surrender, by means of notes attached to arrows fired over the walls. These groups included craftsmen, who would often be sent back up to the Mongolian homelands, and ethno-religious groups - in Baghdad, Christians were invited to surrender their quarter, not because of a preference for the Christian community, but in order to divide and conquer.

Towns were often slaughtered wholesale, occasionally after a perceived slight or token defence. This would be used to convince the next several towns to surrender. Once surrendered, you might be press ganged into the Mongol forces, or simply allowed to continue on as you were, with an administrator or governor appointed, often from the local population who professed loyalty to the regime.

If you didn't surrender though, it was not uncommon for a whole city m, such as Samarkand, to be raised, and the population awarded no quarter. Even then, there is nuance. Official Mongolian accounts suggest that Alamut, home of the assassins, was absolutely destroyed and all its subjects slaughtered, but several notable figures appear in records afterwards, including some, such as the philosopher Nasir al-Din Tusi, who is claimed to have been a 'prisoner' of the assassins, who rose to a prominent position in the Ilkhanid state.

I don't think there is a definitive answer to this, but instead it seemed to vary. David Morgan writes that the Mongol policy was:

That any city that surrendered without fighting would be spared, but that those who caused the Mongol army to suffer casualties would expect no mercy

So theoretically, if your city gave up the ghost before the Mongols settled down to siege it, you would be fine. However, if you actively resisted, the outcome would be different. An example of this was the 1215 capture of Beijing. The Jin had previously held it as their capital, but when the Mongols ravaged northern China they sued for peace. The Mongols left and they moved the capital to Kaifeng, behind the Yellow River. The Mongols came back in September 1214 to the now isolated city. Relief columns were beaten back, and when the city surrendered in 1215, they killed thousands, and parts of the city burned for a month^2

Also, as mentioned further down by /u/Frognosticator, the conquest of Persia was horrifically bloody. Cities that resisted such as Herat and Nishapur apparently have death tolls in the millions^3,4. The sack of Baghdad in 1258 is notorious for the deaths, which range from Hulegu's estimate of 200,000 to the Persian historian Qazwini's estimate of 800,000. This was even after the Caliph had come out and surrendered after a period of siege. Though as /u/JohnGB mentions, all citizens were not treated equally in the sack and Christians were spared due to Hulegu's wife Toghus being a Nestorian Christian herself.^2

There are also examples of cities being peacefully exchanged with no bloodshed. The invasion of Kara Khitai was primarily aimed at ousting the renegade Naiman chief Kuchlug who had persecuted the majority Muslim population, and as such, Jebe announced that under Mongol rule mosques would be reopened and that all Muslims would have religious freedom. Apparently "one city after another" opened their gates to the Mongols and no ill came to them. The capture of the Sung capital Hangzhou in 1276 was a peaceful transition as the Mongols finally captured the royal family. The city of Xiangyang was captured in 1273 after a protracted siege without death after the official in charge of the city switched sides. However, to inspire fear in its' inhabitants, the neighboring city of Fancheng was captured first and 10000 bodies piled up outside the city to be visible from Xiangyang. John Man writes that:

This created a situation the Mongols often faced. To force Xiangyang to surrender, Fancheng had to die, very publicly

So brutality for one city meant on some occasions surrender for the next. Most of the time, if your city surrenders without a fight immediately, you should be fine. Every so often, you end up unlucky.

^1 David Morgan, The Mongols

^2 John Man, The Mongol Empire

^3 Sayfi Harawi, Ta'rikh-nama-i Harat

^4 Juzjani, Tabaqat-i Nasiri

I'm quoting the following part from the history of Jahangushay

(Abubakr, the Fars's Atabak) fought alongside Shah Jalal-al-Din Kharazmshah in Isfahan battle of 624 (1226 AD), but after Jalal-al-Din's loss and observing Mongols' audacity firsthand, returned to Shiraz and asked the two important states of southern Iran, Atabakan of Fars and Gherakhtaean of Kerman to surrender to the rule of Ögedei Khan . . . four years later, he sent his brother with treasured gifts, including a cart full of Persian Gulf's pearls to great khan's Khanserai in Mongolia. He also promised a yearly tribute of thirty thousand dinars . . . Atabak Abubake continued his policy of appeasing mongols along the way, and so did his successors. . . He married his daughter Abash to Hülegü Khan's son . . . (End Quote)

That's the only case I knew of during Mongols' invasions where a city was spared, and it's not all on mongols. Most Emirs, Shahs and Sultans would rather fight to be subjected to the "savage" Mongols. But as a result of that appeasement, Shiraz became a haven for refuges from all over the kingdom and went on to be known as a hotbed of literature, art, and culture.

Good question! People mostly focus on the cities being razed when someone was stupid enough to not surrender. Like most things, the answer is "it depends" but it's not guaranteed to be rosy.

The conquest of the Tatars resulted in the entire military and civilian population being captured and the basic strategy was "kill the leaders and absorb the survivors and all their goods and animals into [the] tribe." If you're a dude, hopefully you were short or young because they also planned "to kill Tatar males taller than the linchpin holding the wheels on a cart" but it's worth noting that the surviving Tatars were "taken in as full members of [the] tribe, not as slaves." Genghis Khan handled this by adopting Tatar children and encouraging intermarriage. If you were skilled in some way, particularly engineers, poets, and musicians, you could very quickly be promoted to a position of power if you intended to help the Mongol military apparatus or enchant members of the court or in conquered cities.

This did change depending on the place being conquered. While plundering Europe, the Mongols were "Disappointed with the material reward of their invasion and eager to show some profit ... the Mongols allowed the Italians to take many of their European prisoners, especially the young ones, to sell as slaves around the Mediterranean." This ended up coming back to bite them when these very slaves were used as soldiers in the Mamluk slave army.

Most importantly, don't be rich and powerful:

"one group suffered the worst fate of those captured. The Mongol captors slaughtered the rich and powerful. Under the chivalrous rules of warfare as practiced in Europe and the Middle East during the Crusades, enemy aristocrats displayed superficial, and often pompous, respect for one another while freely slaughtering common soldiers. Rather than kill their aristocratic enemy on the battlefield , they preferred to capture him as a hostage whom they could ransom back to his family or country. The Mongols did no share this code. To the contrary, they sought to kill all the aristocrats as quickly as possible in order to prevent future wars against them, and Genghis Khan never accepted enemy aristocrats into his army and rarely into his service in any capacity."

Weatherford, Jack. Genghis Khan and the Making of the Modern World