Optifine, as others have mentioned, is a nice mod to have. One of the perks is you can hold down CTRL (or whatever key you bind the command to) to zoom your view - as if you were looking through binoculars. Great for seeing if that thing in the distance is a cow or a zombie.
FastCraft has also been reported to provide significant improvements to speed. It might have a greater effect if you have a bunch of mods loaded, though - not really sure.
Sphax and Faithful are both great texture packs. I've used both of them at one time or another, but I tend to stick with the basic texture pack. More complex textures typically causes more lag, and I play with so many mods, I do what I can to limit lag.
I know you're just getting started, so you shouldn't worry too much about game-changing mods too much at this point. I would recommend you looking at a couple of the minecraft launchers that are available: ATLauncher (atlauncher.com) and FTB (feed-the-beast.com). These launchers provide quick and easy access to modpacks (sets of mods that work well together, and are pre-packaged for ease of use). The ATLauncher contains a modpack called "Vanilla" which is just that - vanilla minecraft, without any additional mods added; however, the Vanilla pack does have an option to enable Optifine very easily, which might be easier than figuring out how to do it manually. There are a number of other modpacks available through ATLauncher that add significant game-changing things (magic, industry, harder monsters, etc) - all of the mods in a pack are clickable to take you to a website that explains what each mod does, so it's a good way to become familiar with them.
There are a bunch of other mods that make minor changes to enhance game play. None of these are required, but depending on how you like to play, may make things more "interesting"...
- Iron Chests: provides a way to make larger storage chests out of iron, gold, and even diamonds.
- Bibliocraft: adds craftable furniture, like tables, chairs, and bookshelves.
- Pam's Harvestcraft: adds a BUNCH of new plant types and recipes to make everything from apple pie to PB&J.
- Inventory Tweaks: a nice client-side mod that will replace broken tools in your hotbar with a spare from your inventory automatically, and various ways to sort your inventory and chests
- NEI (Not Enough Items): when you open your inventory, the right-hand side will show a list of all items in the game. You can click "R" on any of them see the recipe for how to make it. The list is also searchable. Highly recommended!
- Tinker's Construct: make tool parts and assemble them to craft custom tools, like an iron pick with a wooden handle, laced with redstone to provide a speed buff. Somewhat complex, but can be fun.
- Voxel Map / Journeymap: provides a radar in the corner to show your surroundings, and a full-screen map to see a bit further. You can also add waypoints to mark where to find home (or other landmarks).
One of the other mods to look out for is called HQM (Hardcore Quest Mode). This mod adds a "quest book" that provides a bunch of short-term goals that often help guide you through the game or one of the featured mods in the pack.
I know that's a lot to take in, so I'd focus first on Optifine to make things run faster, get your texturepack, and then consider a couple of mods like NEI, Inventory Tweaks, and maybe something for mapping. That's what I'd personally consider the "bare minimum" when it comes to mods.
If you want to play with something significantly more robust, check out the Direwolf20 pack in the FTB launcher, and check out the Direwolf20 Let's Play series on youtube - he does a great job of producing tutorials on different mods and walking through how to use them (his "Let's Play" series is much more focused on "tutorial" than his Forgecraft or "server play" series).
