This is a pretty long and tough level for a beginner. Try to enter this level with an egg because tonguing the egg box as shown in the video is quite difficult and you'll otherwise have to slow down slightly to grab an egg from it.
Next, skipping the beanstalk to enter the first cave area can be done several ways. The way shown in the video is the fastest method, but also the hardest. A simpler method below involves timing a jump such that you hit the Shyguy who jumps out of the pipe while he's in the air, then doing an extended flutter followed by a semi-perfect flutter to enter the cave. This method has very forgiving timing and only takes a little bit of practice. Timing the first jump has a frame window of 5 frames.
An alternative newb-friendly strat involves using the second yellow shyguy to line yourself up for the jump. To do so, eat the first yellow shyguy and fall to the lower platform. Then, you jump and land on the second yellow shyguy. Do a full jump off the yellow shyguy and avoid fluttering. When you land, you should be setup for the jump. Just jump and hold B and you should jump off the green shyguy like normal. Note, you don't do a full jump or flutter during the jump off the second yellow shyguy, you will be misaligned so timing is key!
If you want to see this done, watch the video below.
At the top of the cave, you can tongue the lower of the two red coin shyguys (or bring a different shyguy with you from the lower level) and use him to extended flutter onto the other shyguy, who will then boost you high enough to get over the tall wooden post to the right. This can be used to skip a section in which you become Super Baby Mario, saving a very significant amount of time.
A faster alternative to using a shyguy to get over the post is to use a Toady instead. To do this, you must intentionally get hit near the beginning of the stage and tongue (but not make into an egg) the Toady that comes down to try to steal Baby Mario away from you. The Toady will only show up when you get hit near the Piranha Plants.
As you make your way to the top of the level, you'll likely want to spit out and re-tongue the Toady so that Yoshi does not automatically turn it into an egg before you have a chance to use it. Once you get to the top, upspit the Toady into the pole and then bounce off of it on the rebound, like so:
A somewhat more consistent strat for a new player--only 1.5 seconds slower--involves going below the pipe with the Toady, as illustrated here:
It can be difficult to get the screen to scroll correctly to let you continue moving to the right, however. Yoshi must be low enough for it to work.
The rest of the level is fairly straightforward, though the final cave section can be difficult to do without slowing down. Check the video for reference.
2nd room Comparison
For the beginning of the second room you have 3 different strats to choose from, the video on the right is a comparison between all 3 strats! The top left strat is the fastest but also hardest, while bottom right is slowest but easiest.