I'm kind of new to Arduino and electronics in general. I've played around with a few swtiches and leds and i've been wanting to work with some sound. I've done a lot of looking around, and at the moment, i'm not reading to pay $30 for a shield, so I'm looking into alternatives such as modules and maybe even building one from scratch.
It's not that clear to me either.... I can see how you communicate serially via the RX & TX pins, but I don't see how it connects to the SD card.
If yo look at the specifications, the commands in the table are in hex (hexadecimal =base 16). For example, if you send E8 (232 decimal) over the serial connection, that's a "Volume Up" command. You can use hex directly in your sketch, but if you want to convert between decimal and hex, the calculator that comes with Windows can do it in the "Programming" mode.
There is a serial library for the Arduino for sending/receiving serial data. You just need two I/O pins. Note that the TX and RX connections are criss-crossed. The Arduino's TX (transmit) pin connects to the other device's RX (receive) pin and vice-versa. i.e. You don't talk into another person's mouth, you talk into their ear, and they talk into your ear.
That's usually not practical or economical... Start with VS1053 datasheet. You'd need to have a PCB made and the ability to solder fine-pitch surface mount components... It's not something I can do "at home".
And I haven't looked into what's available, but for a complex chip like this, you'd usually need to buy an evaluation board and probably obtain a software development kit (from the chip manufacturer) in order to do your hardware & software design & development.
How do I use an 8 ohm speaker (resistors, connections, circuit, and most importantly amplification)?]
You need an amplifier to boost the power. Look for a "power amplifier" chip if you want to build one yourself. For example the LM386 is a popular choice as a low-power amplifier. The manufacturer's datasheet will give you the design schematic, showing any additional parts you need to add, etc..