I've been doing some reading on deck building and I'm getting the feeling that using the RtR and GC expansions are better mixed in with a solid M13 deck. However, I can't really confirm this anywhere, which is why I'm asking here.
I've been focusing my booster pack purchases on RtR and GTC but I think I need to spend some more M13 time if this is the case. Any advice?
Generally speaking, whichever option you choose, this is no big deal. Expansions and core sets released together are designed to play well with each other. But core sets are also designed to be reasonable stand-alone products.
Core sets are designed to help you learn Magic. The mix of cards is targeted towards newer players. The creatures and spells are generally about as powerful as the ones in expansions, but many of the lower-rarity cards are simpler. In particular, you're less likely to find surprising or "counterintuitive" mechanics that change the basic resource relationships in the game; cards are also less likely to have difficult requirements or restrictions.
However, there is no one-size-fits-all approach. In my experience, new players learn best when they're interested and excited about the game, regardless of what exactly it is that they're doing. The expansion sets are more complex but not overwhelmingly so. If you're really excited by the idea of taxing your opponent to death, for example, then by all means, grab that (Gatecrash) Orzhov intro deck — there may be a bit more up-front complexity in figuring out how to use the cards well, but your enthusiasm will also help you get up the learning curve faster. My wife learned to play just fine with Myr of Mirrodin, an expansion intro deck; the deck's strategy just clicked naturally for her, so within two or three games she was already making substantive improvements using cards from boosters.
The complexity difference is most noticeable in drafting. Core-set drafts tend to be much more forgiving, with more of a focus on deckbuilding fundamentals like building your deck around a mana curve and choosing cards that support a consistent game plan. Drafting an expansion set well, in contrast, tends to demand a good understanding of the internal synergies and idiosyncracies of the enrivonment.
Cards in expansion sets like Return to Ravnica aren't necessarily any more powerful than cards in the core set. Core sets usually have the "basic" version of an effect, whereas expansion sets tend to feature a modified version that plays to the set's mechanical themes (compare Cancel and Dissipate, for instance). However, there are always at least a few high-powered staples in core sets — cards like Mana Leak, Lightning Bolt, and Farseek — as well as powerhouse "bomb" cards like Sun Titan and Thragtusk.
Tournament decks pretty much universally have cards from a lot of different sets, because being able to cherrypick the best cards available gives players a competitive edge.
One area where the expansion sets really shine is exploring specific themes. Core sets often have some themed cards, like the different Soldiers in M13, but with fewer cards than a full block and less specialized mechanics, they don't go as deep as the expansions do. If you want to build a graveyard-abusing deck, or an elf deck, or an all-artifacts deck based around Tempered Steel, you'll likely want to focus on cards from a particular expansion set that focused on those themes.
If you're not intimidated by the "complexity", than I personally would start with RTR and GTC if you intend to play in standard, because they'll rotate out a year after m13 will
