The warpless route is very straightforward, it just involves a lot of platforming. The first swinging monkey group in the final area can be short-hopped under. The last one can be simply ducked under, or jumped into while simultaneously tonguing backwards. If you do this second option, the monkeys will grab baby mario, but the back tongue will automatically retrieve him. Jumping into the monkeys is about 0.2s faster.
Fast monkey bounce in room 2: 
The warp in 3-1 is triggered by falling above the screen transition area in the trees. Normally, to get to this area, you would take a path down the mountain backwards through the water, and hit a spring arrow that bounces you into a bonus room. If you instead bounce on a monkey and flutter on top of this area, you will trigger the 1-1 warp. It can be tricky getting the monkeys to cooperate, as their movement is highly unpredictable. The IL video here shows a consistent (and fast) way to get the monkeys to behave. Upspit a monkey just after passing the post, then do a full jump while under the first hanging vine in the background. Just hold right and jump to trigger the warp.