It sounds like your at the stage where you need to learn how to recognize imbalances in a position and come up with a plan. I would suggest The Amateurs Mind and any other chess books by Jeremy Silman as well as repeatedly going through tactical puzzle books so that you become adept at spotting tactics in a position. I remember I had trouble understanding idea why people made certain moves or how too come up with a good plan until I read Silman's book "The Amateurs Mind" and then my rating literally shot up from 1300 uscf to 1584 in the course of a few months.
Analysis question
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You're looking for ways to improve. So focus on your biggest mistakes--since you're ~1000 "Long Live Chess" most of those will be tactics. Ask yourself, "Why did I miss such a good move?" Usually it's because: (1) you looked but failed to see it (you need to practice similar tactical patterns) or (2) you didn't look (you need to work on your thought process). (3) Often psychology plays a role (you looked for ten moves straight but then forgot to look because x, y, and z happened). Engines help with tactics, the game explorer helps with openings, and tablebases help with endgames. Coaches and other players can also help. :)
I usually write out my feelings, evaluations, and calculations before an engine gets involved, so I can find flaws not only in the moves played, but also in the moves that might have been played and my general evaluation skills.
Analyzing your games with a chess engine would help you improve.
By this I don't mean plugging your game in the engine and then coming back later and looking shortly at the analysis. I mean using the engine to understand why a move was bad or why one move is better then another, and this requieres quite a bit of work.
You all mention chess engines. How do they work? Do they show percentages of success, or will they show alternate lines? And to get a little further down the rabbit hole, which ones are good, and what do they cost?
In a particular position, a chess engine will do two things for you: (1) evaluate it, and (2) list the top few moves. For example, after 1.g4 f5 Fritz says:
1. ± (0.81): 2.gxf5 d5 3.e4 Qd6 4.d3 Nf6 5.Bg2 dxe4 6.dxe4 e6
2. = (-0.19): 2.g5 e5 3.d4 exd4 4.Qxd4 Nc6 5.Qe3+ Be7 6.Bg2 d5
3. = (-0.19): 2.e3 d5 3.gxf5
In other words, if White plays 2.gxf5 he'll be winning by about a pawn (+0.81 pawns, technically). If he fails to capture, then the game will be about equal (he'll be down by 0.19 pawns, technically). It will make sense as you get used to it.
If you have a PC, the top engine is Rybka. It comes in three flavors: 3000 elo for free, 3150 elo for $50, and 3200 elo for $100. The pay versions come with a Chessbase or Convekta user-interface. If you get the free Rybka, you can get a free user-interface online called Arena. It's not as slick as the other two, but it's quite capabale.
A last option is Chessmaster. Its engine is "only" 2750 elo, but the user-interface is extremely easy to use, comes with a game database, and comes with many tutorials. So if you want more than just an engine, that's a great pick. :)
Another thing you can do with a user-interface & engine, usually overnight, is ask to to evaluate all of the positions of your game and report your biggest mistakes. I rarely use it in this mode, but some people do, so I should mention it.
It’s been said many times that game analysis can help players improve. I’ve been playing for a little over a year, and as silly as it sounds, is there any advice people are willing to offer regarding personal analysis. At the moment, I look at my own games and simply try to find my errors, and I can usually pinpoint them, however, I have a hard time trying to find what the right move/line should be. Obviously, there's much more to it, but as someone new to the game, should i be doing anything else in my analysis?