Seeing as you didn't mention what your personality is like, we have a tough time thinking of what sort of decks would suit you. But since you're starting out again after so long, I'll just give you a brief rundown of things:
1) Synchro Monsters are a new type of monster that acts like a cross between ritual and fusion monsters: they are special summoned from the extra deck (what used to be the fusion deck) by "tuning" a tuner monster with x number of other monsters that equal to the synchro monster's level. The some of the more popular ones are: Black Rose Dragon, Stardust Dragon, Brionac, Dragon of the Ice Barrier, Mist Wurm, Ally of Justice Catastor and Goyo Guardian.
2) The current top-tier decks are: X-sabers, Gladiator Beasts, Blackwings, Machina Gadgets, and Lightsworns. Out of these, the easiest (and cheapest) to build would be Machina Gadgets and Gladiator Beasts, and the most expensive would be X-Sabers.
3) Structure Deck wise, there are a couple right now that worth getting:
*** Machina Mayhem is a machine based deck, and getting 3 of these would give you the basic structure for a tournament winning Machina Gadget deck. Key cards in this structure deck include Machina Gearframe, Machina Peacekeeper and Machina Fortress. Fortress is a 2500 atk powerhouse with the option of being special summoned from the hand or grave simply by discarding machine monsters who's levels equal to 8 (made easy with Fortress himself being a lvl 7). Fortress also has a number of effects that activate depending on how he is destroyed, making him a pain to get rid off. Gearframe can search for any "Machina" monster in your deck when it is summoned, and is a union monster that prevents the monster it equips itself to from being destroyed (once). Peacekeeper searches your deck for a union monster when it is destroyed, so you could use it to search for Gearframe, then summon Gearframe next turn and search Fortress.
*** Structure Deck Marik is the latest structure deck which is based on Marik Ishtar's cards from the anime, as well as Gravekeepers. The Gravekeeper archetype revolves around the field spell Necrovalley, and typically focuses on locking your opponent's graveyard while you exploit loopholes to abuse your own. But the best part of this deck is the fact that it comes with Mirror Force, Allure of Darkness and Foolish Burial. Mirror Force destroys all your opponent's atk position monsters, Allure is an incredible draw engine for Dark decks, and Foolish is a great card for decks that abuse the graveyard.
*** Zombie World is a nice deck that would help classic players get back on their feet. Featuring a unique spell card called Zombie World (surprise surprise), it turns all monsters in the graves and field to zombies. This can potentially lock down your opponent, especially if they are using cards that rely on a specific type (i.e. Marauding Captain can no longer protect itself because it is now a Zombie). It also has a variety of cards that take advantage of this situation, like Red-Eyes Zombie Dragon, which special summons a zombie it destroys by battle to your side of the field.
If you want a full list of structure decks, I suggest going to yugioh wikia to check which structure deck suits your tastes.