Reddit mentions of Premium Dungeons & Dragons 3.5 Dungeon Master's Guide with Errata

What you've got pictured in that first link is the 3.0 Player's Handbook. (There is no such thing as 2.5, despite someone's mislabeling.) For comparison, here's the 3.5 Player's Handbook which looks rather similar, but has "v.3.5" near the bottom.

Regarding visual tools (playmat, miniatures, etc), those are often helpful, especially for new players/DM's and for those who like tactics-heavy battle, but I'll note that they aren't required, and some DM's eschew them on purpose, focusing instead on vocal descriptions of events.

Do you have a Dungeon Master's Guide? Here's a 3.5 DMG - the 3.0 and 3.5 versions are close enough to work together without too much trouble, and 3.5 is generally seen as superior, but a 3.0 DMG would work too. The DMG has a wealth of information useful for running an engaging game.

If you're, say, missing some key element of DM'ing, I'm not sure we could identify it easily - if you can't see it, we can't tell you about it. In general, though, it helps to try to pay attention to what does interest and engage your players, and do more of that. Or if some part seems to annoy a player, find a way to make it less of an issue.

One thing some players like, is to feel like they have some special trick or advantage they've figured out that gives them a special advantage. Maybe your fighter would think making a trip attack or sundering a weapon would be an awesome thing to do. Look for opportunities to let him try unusual stuff when it'll have dramatic effect.

Note that players can have radically different types of games that interest them; there is no magic bullet here. The original D&D had characters with short lifespans, who were basically just out looking for loot and had backstories like "merchant's disowned son looking for riches" at most. Some games have virtually no combat at all, focusing entirely on verbal descriptions. Some players enjoy basically enjoying going down a predefined story, and others like a more "sandbox"-like experience where they choose where they go in the world.

Another thing that can help is getting some practice. Try running through some combat scenarios on your own, doing all rolls and decision making for all (2-6) characters in combat. This will help you get used to the flow of combat, and let you handle it more smoothly in a real game. It won't help you with other skills, like interacting with players and reacting to their unexpected decisions and goals, but confidence here will help elsewhere. You can focus on the goblin chieftan's speech more if you aren't worried so much about handling a pair of attack of opportunity.

If you feel you're having trouble with storytelling, it might help to use a "module" - a pre-made setting and story, with some ready-made adventurers and lots of flavor to follow. If you do, you don't have to follow the module exactly (your players almost certainly won't!), but having the structure to rely on will serve you well.

I'm going to disagree on your point of editions. 3.5 is still massive, even with Pathfinder eating into it. WotC just recently released a new printing of the PHB, DMG, and MM1. There's many out there who still play this and have a trove of these books. I've played in several groups over the past 5 years, and they all played 3.5*. That said, most 3.5 stuff could probably be covered at the same time as Pathfinder.

^(*And I never sought out any specific system when joining. I've actually been itching to try out Pathfinder, Savage Worlds, Call of Cthulhu, Shadowrun, the new Marvel RPG, and just the other day backed the kickstarter for "OVA: The Anime RPG". Too bad I only have time to be in one group!)

If you're completely new to DnD, I'd go for either DnD 3.5 or Pathfinder. Neal uses an apparently very homebrewed 2nd Edition, so, you're not going to be playing the same game with regular old 2e stuff. It's clunky and aged, and he (presumably) still uses it because it's what he's always used, so he's used to it, and he's ironed out the kinks.

So, yeah, the most recent edition of DnD is 4e, which is commonly considered to be a little bit too dumbed down, but it doesn't really matter that much if you're new. Still, the books are expensive, so you should try to look into the different versions and find out which one sounds like you'd prefer it. 3.5 and Pathfinder are very similar, as Pathfinder is actually based on 3.5 with a couple things done to streamline some of the combat. Also,the art in the books is really pretty (completely superficial, but it's nice,) and you don't have to buy a rule book, player handbook, AND a bestiary, just the core book and a bestiary if you're DMing, so it's like thirty bucks cheaper, AND Pathfinder is backwards compatible with 3.5 books, so... yeah, that's also nice. They also have a DM's guide which is all about how to make and run a game. Pathfinder and 3.5 both have a bunch of ancillary resource books too, like NPC and equipment books. I know Neal has at least one weapons book, but they aren't necessary; they're just extra guidance.

>I know how hard it can be to get your hands on these

Not that hard. amazon.com is selling them for under $33 each (retail $50), and they qualify for Prime shipping:

• Player's Handbook
  
• Dungeon Master's Guide
  
• Monster Manual

Oh yeah?