I own Lingua Latina. It has an immersive method, where you start reading everything in Latin. If you want a grammar based traditional textbook, Wheelock's is the classic one - and it has good online resources. Teach Yourself Latin is good, except I find the binding bad - it started falling apart pretty easy. When learning Latin, or any language (at least one where you are mostly going to be reading), I recommend paying special attention to translating from English to Latin. This really helps build your vocab and knowledge of grammar.

I concur with Sayers in the essay I linked to above, learning a highly inflected language like Latin really does improve your grammar and your understanding of language and even thought. There are many such languages you could learn - Russian and many of the Slavic languages, Old English, Old Norse/Icelandic, Irish and Old Irish, Gaelic, Basque, Sanskrit, Arabic, Korean, Japanese - to name a few. But for the Westerner, especially one interested in historical literature, culture, and philosophy, Latin and Greek are obvious choices. Both are hard in their own ways - Classical Latin is very complex, whereas ancient Greek, though I would say less complex than Latin (even in the older forms like Attic or Homeric) obviously uses a different alphabet to English and also has a slightly annoying accent system to remember.

I would also recommend Peter Kreeft's Socratc Logic as a good introduction to logic from a more Aristotelian and traditional perspective.

Yes, Lingua Latina is certainly an easier entry into Latin. They do say that your reading skills will be slower if you use a grammer based approach. However, the main aim for most in learning Latin is to read the classic authors - Cicero, Caesar, Tacitus, etc - and won't be able to read these simply by reading Lingua Latina, not without looking a lot of things up. The same goes for Wheelock's actually. Completely it once won't allow you to read the Cicero readings at the back at anything but a very slow speed.

It is accounted for, in the sense you do more or less learn the constructions required. It is just the literary style of an author like Cicero is very advanced and you have to decipher it. I can read, after a year of Latin, modern works translated into Latin, like The Hobbit or Harry Potter with moderate ease (despite the fact the latter uses quite a few strange constructions and terms to describe modern or magical things), but I haven't even bothered yet to buy or look up Cicero's works in Latin, other than what I have in Wheelock's reader and the readings from the textbook I own, as I am not up to reading them in full yet.

Latin is also a relatively vocabulary poor language, unlike modern English. This means they have to get the most out of fewer words. It also means they use many idiomatic constructions that take some getting used to.

Early on I skipped a lot of the translating of sentences from English to Latin, as I thought why would I need to do that. This was a big mistake. I think this probably teaches you even more than translating from Latin to English.

One resource I've really liked so far is Peter Adamson's History of Philosophy Without Any Gaps podcast. A new episode every week for most of the year, with the intent to cover the whole history of philosophy! He's at the thirteenth century now, which hopefully means a whole host of content soon relevant to the topics discussed on this forum. (Caveat lector, I've only listened to about 100 episodes.)