It tends to vary, and both examples can be seen. Early on, after the new formation of the Mongol nation under Genghis Khan, the nation of Hsi-Hsia (or Western Liao dynasty) was attacked. Initially there was a pitched battle where the Mongols faced an army of 50000 men and won^1. These were Chinese armies of infantry combined with cavalry, which the Mongols initially beat and were then halted for two months whilst dealing with a second army.
In the subsequent war against the Jin/Chin, the Mongols broke behind enemy lines due to help from the Onguts, ethnic relatives to the Mongols who guarded the border. The Mongols picked the battleground (battle of Shansi) and engaged the Jin in a set-piece battle, whereby the Jin were utterly defeated when Subotai arrived from the rear and attacked the Jin flanks.
In 1218 the Mongols moved against the Kwarizmian Empire, where initially one pitched battle was fought between Jebe and the shah of that Empire, which Jebe won. The Shah then refused to meet the Mongols in the field and garrisoned the cities such as Bukhara and Samarkand. Unfortunately for him, the Mongol army was split up and came upon Samarkand from multiple directions, encircling his forces and cutting them off from the southern portion of the Empire. The Shah fled and with the armies locked up inside the cities the Mongols effectively held Transoxiana without much of a fight. They then proceeded to take Afghanistan and Khorassan, and whilst the Shah's son raised a new army it was destroyed in 1221 on the Indus river.
Then Subotai launched his cavalry raid. He met the army of Georgia in a pitched battle, where they feigned retreat whilst tiring the Christian army. The Mongols then mounted fresh mounts located at the rear of the field of battle and crushed the army of 70000 men. A second battle against the Georgians in the winter utterly destroyed their army. After crossing the Caucasus a coalition of peoples met the Mongols and forced them into a defensive position, but they bribed a portion of them, the Cumans, into leaving the battle. They then destroyed the remaining armies and later the Cumans themselves.
Following this, they advanced into Russia, where Russian princes met them and initially pushed them back. They followed them for nine days to the Kalka river, at which point the Russians were spread out and tired. The Mongols met them in the field and destroyed the front lines of the Russians, met armies who were following the initial princes armies and crushed them too. The Prince of Kiev and remaining Russians fled the field, where the Mongols chased them down and killed them.
Following this, after the death of Genghis and rise of Ogedei, the Russians were attacked again in 1237. The Mongols destroyed cities like Ryazan and Yaroslavl. The Russians joined up on the Sita River, where they were surrounded in the snow and crushed. The subsequent campaign against Hungary involved operations protecting the northern front in Poland, where the Mongols slaughtered the Poles and Teutons at the battle of Leignitz^3. In Hungary, Bela IV advanced to meet the Mongols till they retreated behind the Sajo river. The Hungarians initially held their ground till a flanking attack by Subotai who emerged surprisingly from the south, at this point the Hungarian camp was encircled, and as they fled they were destroyed. Advancements after this were prohibited due to Ogedei dying, and the long pause that followed.
Eventual conquests such as Baghdad by Hulegu^2 met little opposition before the city was sieged and taken. Hulegu moved onto Syria where Damascus was abandoned to the Mongols in 1259, but the death of the current Khan Monkhe led to him taking two thirds of his forces home. The remaining 20000 famously lost to the Mamluks at the battle of Ain-Jalut.
Sometimes however there was little opposition, with the Kara-Khitai falling relatively peacefully into Genghis Khan's hands early on in the 1210's, and fear compelling the surrender of other cities. Other times such as the capture of Xiangyang, military officials were pursuaded to change sides if cities fell over peacefully. The Mongols achieved their conquests through both destroying armies and terrorising cities open to them.
This is quite a roundabout answer but I hope it's useful.
^1 Genghis Khan's Greatest General, Subotai the Valiant, Richard. A. Gabriel
^2 The Mongol Empire, John Man
^3 The Mongols, David Morgan