Simply put, you have to get all the in-game badges to either Gold or Platinum rank. You do not need the 'Elite' badges for this achievement.
Points Based Badges:
-Bejewler badge: This one shouldn't cause too much trouble. Keep up a combo and gather special gems to make a 'spectacular' move to get more points. If you can survive until round 10, you should be able to get this one easily.
-High Voltage badge: This one is the easiest badge to get if you're used to playing Bejewelled Blitz. All you need for this one is speed. Make lots of combos really fast and get as many of the time bonus gems as possible (although, when going for score, I wouldn't recommend focusing on them too much). Us the LB button for a hint if you don't see a move immediately, there is no penalty for it.
-Ante Up badge: Get as many Full Houses and 4 Of A Kinds as you can to get this badge to gold. As long as you try to avoid getting a pair too often, you can make a Poker game last for quite a long time, giving you plenty to get to 500,000 points.
-Glacial Explorer badge: For Ice Storm, I recommend using the same strategy as for Lightning. Make lots of combos really fast. I got a platinum in this without paying attention to the ice in the background, I find that trying to stop certain ice columns became distracting and I wasn't getting very high scores or combos.The Star and Flame gems are your best friends in this game mode as they knock down more than row.
-Diamond, Mine badge: The main aim with Diamond Mine is to make all your matches at the bottom. Star gems clear a lot here, but fire gems are also really useful. If you can make it down to about 200 (on the number on the drills to the side) you should be able to get this with no trouble.
-Butterfly Monarch badge: Butterfly mode has no timer, so you can plan your moves. Apart from trying to get a butterfly with every move, but also using special gems to clear butterflies, I can't think of much strategy to this one. See what butterflies are where, and make each move count for something.
-Final Frenzy badge: This one is easier than is sounds. Play like normal, but keep getting 4/5 of a kinds, and then not matching the super gems they leave behind. Having about 7 super gems blowing up in the final frenzy (assuming they are not all fire gems) should be enough to get gold for this badge.
-Ice Breaker badge: Speed. Just be really really fast and have some Star/hypercube gems at the ready. you keep comboing these, and you'll get it in no time.
-Relic Hunter badge: Much harder to get Gold in this than it is to get Platinum in the points on for this game mode. If you're lucky, you can get this down to 180. If not, you will be digging to nearly 300 before you find 12 relics needed for the gold badge.No real strategy except for survive as long as you can.
-Butterfly Bonanza badge: If you can make a supernova gem, this is easy. Or, 2 hypercubes. A single hypercube would be fine in a combo. A fire gem can get rid of up to 7 butterflies in one explosion. Make a small chain of a couple of super-gems and this one should be yours. Obviously wait till later in the game to activate it, when there are more butterflies to catch.
-The Gambler: Now this one is a pain in the *ss. I'll attach a video to the bottom of this showing you a neat technique to get them faster, but it's still a really slow method, there is no really fast method :/ In the video, he's on the PC version, so there is a reset button. For us Xbox users, we just press the 'Back' button to restart a round (Thanx to demonWhistler for letting me know you can press the 'Back' button to restart a round). It keeps all Flushes recorded even if you quit out and start a new round. (I've been trying the method in the video with quitting out instead of restarting, and it's doesn't work all the time. So, a few pointers: Hypercubes. Any hypercube you make gives the card a '?', meaning it can be any colour. you then also have a hypercube which can get you any colour combo with something it touches. Make 3 of the same colour combo, a hypercube and then link that hypercube with another of the same colour as the previous 3 for an easy Flush. Of course, 5ofakinds don't come around that often :/ )
One important note for this: You must finish a round to get a badge. So, when you get 10 Flushes for the Bronze, finish a round. When you get 20 for Silver, finish a round etc. You can check how close you are on the badges section. if you go over, it will say '21/20' for example. Go finish a round and all the excess flushes you got will add to the next amount you need.
-Blaster badge: There are several ways of going about this. The easiest is to make a Supernova gem. But, a supernova gem itself is not enough, you would have to make some Star gems or fire gems that would also chain react with the supernova gem when you set it off. The other way you could do this is with Hypercubes. if you have a hypercube touching another hypercube, you can react one to the other to clear the whole board. The only time I've done this is in Diamond Mine, so I don't know if this will get you the badge or not, but it's worth a try (it will get you an Elite badge as well).
Inferno/Stellar/Chromatic badges: Getting a lot of super-gems will take a while. If you have all the other badges, I recommend trying this on Lightning mode. While you may not make as many super-gems as you would in say classic mode, all the super-gems left at the end will explode and count towards your badge. Going by 'super-gems destroyed per minute', Lightning mode is definitely your best bet. Other people seem to think Diamond Mine is also a good bet for the Hypercubes and Star gems. It is true you get more hypercubes in Diamond Mine, so if that is the last one that needs Gold, I'd recommend going there also.