To Warp, carry a Potion Vial to a Jar that you can't normally enter. In Sub-Space, you can go down the jar for a warp.
 

World 1:
OK, the locations of the Mushrooms through all this world place them in about the same space you find the potion vials. Exception is in World 1-2, where the vial on the Shy Guy's grass is used solely for Coin collecting from the weeds to the right.

1-2: If you get a Starman on top of the grassy outcrop with the Gray Shy Guy, get it as close to the door as you can. You'll be able to blaze through Birdo if you RUN!

1-3: Get all the Cherries, but DON'T get the Chery near the Mushroom in the first Sub-Space, unless you're POSITIVE that you can grab the Starman and the Mushroom swiftly.
 

World 2:
This desert has a LOT of quicksand and digging sand. If you can, get a Starman while digging, it makes life MUCH easier.

2-2: The Mushroom and 1-up are located in the first door in the level. Blast the walls with Bombs to open up a clear path.

2-3: To enter the high door at the start of the level, hop on a Beezo, then hop again. But first, jump to make them fly high enough for you to jump from the Beezo to the ledge, but low enough to jump on them to begin with. 
     Also, note that the Mushroom is left of the potion vial, jut carry it over to where the Cherry is, left of the ladder.
 

World 3:
First a vertical climb, then a partly underround adventure, then Mouser's tower of terror. It will become a tower of error when you get through with the rowdy rodent.

3-1: If you think going up is tough, try going down. You'll land above a door leading to a MULTITUDE of grass clumps that will become precious Coins in Sub-Space. Look for the potion in the 5th or 6th clump to the left of the jar at the end of this room. Remember, you can weed for Coins twice in ONE LEVEL, so if you weed like crazy twic in World 3-1, then warp to world 5, you'll get Coins when you do some Sub-Space gardening there as well, and have a LOAD of Coins for the slots.
     If you chose to go up instead, stay near the center of the screen, and hold "UP" after tossing Pidgit off his carpet, that way you can grab the vine easily.

3-2: Use Luigi or Princess to super jump over the wide gap underground, and
skip some unwanted action above

3-3: The best shortcut is to carry Ninji from the bottom floor of the room he's in to the last ledge you can jump to. Put Ninji down and stand on his head. Now super jump to the ledge above. Other than that, just try not to get burned or zapped or hit.
 

World 4: The Mario troupe crossed the desert, now they must break the ice.  Flurries in this region strike hard, and by that I mean the quick moving enemy on ice skates. If you're quick, toss them! You can warp here from the jar at the end of the outdoor part of World 1-3.

4-1: The Mushroom sits to the left of the first potion vial, but the coinage is to the right. To get to the Rocket Ship, run out on the ice to the very edge, then hit Left. You'll fall as opposed to turning (this IS ice), but you'll land on the rock with the rocket. Oddly enough, there is no Birdo in the last screen, but lots of Autobombs and Flurries.

4-2: To dodge the Beezos, just run, slide, and duck, and throw in a jump when necessary. If you want a Mushroom, make a risky grab for it on the tail of the whale to the left. For coins, carry the potion to the right. You'll find the potion on the whale's back. CAUTION!!! Their spouts hurt, and could knock you backwards. Again, go for a Starman close to the door on the screen with the spikes and Autobomb. You'll be able to skate past Birdo if you're swift.
     If you'd rather warp to World 6, carry the Potion Vial past all the Cherries, over to a ledge with a jar on it, located under the ice bridge with the Flurry on it. Enterr the jar in Sub-Space.

4-3: Get a Potion Vial at the start, and carry it left for a Mushroom. To kill the Birdo here, hit him with either two eggs, or an egg and one of the Vegetables. Pull up the other Vegetable for your last hit, but stand to Birdo's right and jump on an egg before tossing the Veggie. DO NOT hit any buttons until the egg carries you over to solid ground. To skip the twin towers, have Princess or Luigi super jump over the gap between them.
 

World 5: THIS was a strange, dark place, and NO ONE in Mario's group liked it. They could have warped here from World 3-1.

5-1: Princess Toadstool is the best choice for tackling this level, as her jumping and floating skills prove vital for crossing the gaps, especially if Trouter's your lifeline. To get the second Mushroom, jump off the log, over the wall, and across the first gap. Go down the second gap to get to the
Mushroom.

5-2: There's a Potion Vial down a jar, but to get it, you must blast the floor. Since you can't use it in the jar, carry it outside, and toss it on the ground to the right of the jar. The Mushroom will be on lower ground.
     There's no trick to dodging the spikes, just drift away from them as you see them coming.
 
5-3: If you can get to the hight ledge with the jar with a Potion Vial, you're World 7 bound. Otherwise, there are no tricks here, except to blast any walls you can in case of nearby Mushrooms from Sub-Space. And carry a Mushroom Block to deal with the two Pansers that assault you as you climb inside the tree.
 

World 6: Another desert, but most of the sand is fast.

6-1: In the first jar, there's a Potion Vial. Carry it outside, and toss it on stable ground. The Mushroom will fall into the qucksind, so grab it fast!
     Remember that the Potion on 3-1 was in the 5th or 6th clump of grass?  Well, the jar with the Key and Phanto is the 5th or 6th from the far right door. Also, toss Cobrat, so he rams into jars with Mushroom Blocks on them, that way he'll kill the Cobrats inside. Down those jars are sand bars with a 1-up at the bottom, but if you get one from either side of either room, is negates the others, turning them into small veggies.
 
6-2: Nothing special, just ride the Albatoss to the right, climb onto the highest flying one going left, then jump to a high-flying one going right.  Get off to get the Potion and Mushroom where you see grass growing, and then go back and ride the high flying Albatoss to the end. 
 
6-3: Sink into the sand left of the start, and once you're below the level of the mountain, jump while holding left to go under it. Keep jumping and hold left until you find yourself by a door leading to a point near the level's end.
     If you travel the cave, let the Bob-Ombs blow up their own jars while you bag Cherries. Blast only some of the lower walls to find the Mushroom, otherwise you'll be forced to exit without it. As you climb the vines beyond, notice that the leftmost vines usually carry you up further.
 

World 7: Wart's place, and it's HARD. Have fun, and GOOD LUCK!

7-1: No real tricks, other than to find a Mushroom on the ledge near the center of the screen with the Albatosses, or to pile up Mushroom Blocks in the clouds and super jump up. To ease your pain, try to get a Stopwatch in Albatossville, or a Starman after you get the Rocket. To get that, ride an Albatoss to the left, and jump over the high poles.

7-2: OK, this is so complicated, I made it a sort of a walkthrough. First, go as far right as you can, and go up the chains and through the Sparks. When you get to a fork in the road, go up the chain on the left, and out the door. Below are 2 Tweeters and a Panser. in the room below, get a Mushroom, but don't toss the Potion Vial in front of the main door, or you'll end up outside and have to enter again. Go back out after you have three HP, and get back to the main hall where you first began climbing up.

Now, go down the chain under the conveyor belts. The next rooms are in a sequential order, so I've numbered them. Note that if the screen scrolls with you instead of switching to another screen, you're not in a different room.

Room #1: Stay near the center as you pull the Mushroom Blocks out from under you. That way, you won't end up falling on the wrong side of the ledge if a wall separates the ledge into two sides. There are Cherries on these ledges, and you'll need them. 

Room #2: You should be invincible now, so crash through the Sparks as you go right.

Room 2a: The door in Room 2 leads to a Potion Vial and Sub-Space. Grab a Mushroom here to increase your life meter to 4.

Room #3: You'll climb some chains, with Sparks and Cherries on the sides. Since your invincibility would wear off before you reached Room #4, don't bother collecting.

Room #4: Conveyor belts, Sparks, Shyguys, and Snifits serve to make this vertical room into a test of your jumping skills. Toss whatever you can lift into something else.

Room #5: Birdo waits here, and ther'es a conveyor belt and a low ceiling. Run off the chain so you're on the ground below the conveyor belt, and run under Birdo to avoid a tough fight.

Room #6: If you took the path to the right (see paragraph above for first Mushroom), you'd end up to the far left of this spot, and above it at that.

Room #7: Birdo holds a Key here. Beat him, and take it, then go back to Room #6.

Room #8: Use the Key to get here, grab the Crystal, and fight Hawkmouth. Go inside after three hits.

Room# 9: Wart's chamber. Gulp...