EVE allows you to stay offline and your skill training will still progress. But is there much point in paying a sub if one has little time to play and already many other games to play?
You could do this in a later stage when you are not only able to fit and use battleship, but to fit and use them and all other ship classes really well.

Early on one must really play for quite a while to get into the game, do missions for an agent, mine some ore, all that.

I wonder what you are supposed to do in 0.0 systems. 0.5 is good enough, and 0.1-0.4 do not really offer so much more than more ore, more dangerous rats (enemy npc ships) and some more dangerous complexes. I read you trained enough to use a destroyer by now. I.e. you can now use it and fit some guns on it, but your training level in general is still rather low. You simply cannot take on the more dangerous mobs right now. But it would pay off to run the early level 1-2 missions so that you can soon tackle the more interesting level 3 missions afterwards.

I would not enter 0.0 unless a fleet mate takes you by the hand, and even then, you have no business to do there. You would rather fall victim to rats, pirates or most likely a gate camp.

You could of course stay subscribed for some 6 more months and then be trained really well, but I think you must explore yourself much earlier what keeps you enticed to play this game. I was mostly running plexes and doing missions, occasionally playing escort for buddies doing mining operations and ratting in low sec space. Some others focus on the “industrial route” an do mining, but this alone would have been too boring for me. I was rather unfocused, Jack of all Trades, but I liked that. I did not specialize too much, I had cruisers, battleships, frigates and industrials of various races trained really high in the end.

One of my ambitions in EVE was to get a NIGHTMARE battleship.
One of my buddies even got a Machariel.
http://wiki.eveonline.com/en/wiki/Nightmare