In Episode 110-120 I think. During the Arabasta scenario.
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It depends on which version you watch. In the 4kids tv dub, Luffy chases Crocodile down into some ancient ruins and then beats him up because he sweats, which is water, thus allowing Luffy to beat up Crocodile. In the Raw version, its the same scenario, except it is because Luffy is bleeding so much.
because she loves him
At the battle of Marineford, it was shown that Kizaru was still easily faster than Luffy's Gear Second speed. It is not known if Luffy's speed has increased further after the timeskip.
The first one Luffy defeated was the marines who took Roronoa Zorro to be executed. He went on to his adventure with Zorro and met Nami and they defeated Arlong, the sharkman who conquered Nami's hometown. The most remarkable opponents he defeated were Crocodile Gecko Moria and Eneru. They were very powerful that they almost killed Luffy but Luffy succeeded. The major (and minor ones worth mentioning) characters he has defeated; - Helmeppo, - Alvida, - Buggy the Clown, - Captain Kuro, - Arlong, - Don Krieg, - Wapol, - Mr. 3, - Sir Crocodile, - Bellamy the Hyena, - Enel, - Foxxy the Silver Fox, - Blueno, - Rob Lucci, - Oars, - Gecko Moria, - Monkey D. Garp, - Hody Jones.
Luffy well become the pirate king after beating the 4 emperors, That mean he need to beat big mom, shanks, black beard and Kaido.