Ruby and Sapphire post game stuff
Sky Pillar, located on route 131, is home to a level 70 Rayquaza (note that you can get Rayquaza before the League in Emerald). However,reaching the top is tricky because of the cracked floor, so you will need a mach bike.
First floor: normal
This floor shouldn't be too hard
Basically nothing, just where you land.
This is where things get tricky. You'll want to be careful, but make sure to keep going fast. It may take a couple tries but you will get there.
*Fifth floor: normal
After defeating the champion, Latios(R) and Latias(S) will be roaming around the region. In Emerald, there is a report, and the Lati that roams is based off of the color you pick.
Battle tower(RS), Battle frontier(E), and S.S. Tidal
After beating the champion, you will receive a S.S. Ticket to board the S.S. Tidal. When going to the Battle Tower (RS), there will be various trainers you can battle and items you can find. In Emerald, you can go to the Battle Frontier, which has the Battle Tower, Battle Arena, Battle Factory, Battle Pike, Battle Dome, Battle Palace and Battle Pyramid, each with have a Frontier Brain. There are also move tutors, Artisan Cave, trades, and Sudowoodo.
Located on route 111, north of Mauville, it will be open after entering the hall of fame. It has plenty of trainers to fight, and rare prize items.
The final puzzle is unlocked after defeating the champion. The prize is Red tent(R), Blue tent(S), or both for Emerald. These are for your secret base.
In Emerald, there is an add on to the Safari Zone which has some rare Pokemon.
In the back of the Fossil Maniac's place, there is a cave which has Ditto and the Fossil you didn't already get.
Steven and Wally(E)
In a room in Meteor Falls, you can battle Steven (since Wallace is the champion in Emerald) You can also battle Wally at the end of Victory Road.
Gym Leader rematches(E)
Using the Pokenav, you can have rematches with the gym leaders.
Groudon and Kyogre(E)
Groudon and Kyogre can be found in the Terra (Groudon) and Marine (Kyogre) caves. Talk to a person at the Weather Institute to know which routes the caves are on. The caves randomly appear and disappear, and only one can be present at a time.
Pokedex Update and Johto starters (E)
After doing the hall of fame, your Pokedex gets updated to the national dex, and after getting all 200 non-mythical Pokemon from the Hoenn dex, you get the choice of one of the Johto starters.
After going into the hall of fame, Altering cave will be on route 103.
You can get the phrases CONTEST, SUPER, LATIOS, and LATIAS after you go through the hall of fame and see Latios and Latias. These phrases can help with getting rare berries from the Berry Master's wife.
Source: Bulbapedia and experience. Let me know if I missed anything or if anything is wrong. Hope this helped.