If you really want to be "competitive", especially when you reach "end game", you will have to be prepared to spend real cash hard and furious. Even with the "new trend" of "events" that offer massive boni to gear upgrade rates (e.g. +60% and the like), it would still take effort and money to create the "perfect gear".
It is always possible to be a "free player", but it will take just forever, although again, it is much easier these days to level and farm resources with all the "events" mentioned above, even for "free players". But when it comes down to the "uber" gear, it is either you will have to spend your own money, or, if you do have friends that could provide you with the needed mall items for lants, you would have to farm endlessly and acquire mountains of lants (and resources).
Most of the leveling can be done solo, although partying always speeds up the process, especially at higher levels. Most dungeons / instances require a party, and the high level ones, a good party and good gear, otherwise it will be very lengthy fights and probably fails ./ wipes too. End-game dungeons like ET (Erunil's Tower) and AoS (Altar of Sacrifice) are seriously hard and probably impossible to do for the "average" geared player / party. Most players running them efficiently are all in top, heavily upgraded gear only.
Finally, about the Premium subscription, it is not required, you can buy it, but the benefits are not that very great anymore, with all the numerous and frequent events floating around. Premium would probably help more in the beginning and through the leveling process, rather than later on and end-game.
I don't know, personally, I can no longer recommend Requiem to new players, there are so many far better options out there. If you do want to try though, give it a go, but as I said earlier, better don't be tempted to start spending until you have decided, and also, if possible, try to join a decent guild that doesn't mind helping new players. The population these days isn't really high, so depending on "randoms" for parties of any kind isn't a very good idea.