1) 60 cards and an opponent.
2) Learning which cards are better is the experience of the game. You can read up or you can play. There are differences in rarity. This comes in two ways. In each set or printing, cards have a mark about half-way down the right-hand side of the card, under the art. The color of this mark can be black (common), silver (uncommon), gold (rare), orange-red (mythic rare). In addition, some cards get printed in more than one set, so this results in more of them available in the market.
3) I store my cards in card boxes. Personally, I put sleeves on my cards that are worth more than a buck, but you're free to do as you like. For cards in a deck, sleeving is advisable for the entire deck.
4) Depends on the format you're playing. If you're playing "standard," that means only certain recent sets can be played. Which sets those are changes over time, as new sets rotate in, older ones rotate out. If you're playing in a legacy format, or kitchen-table style, you can play whatever you have.
5) The top right shows the casting cost of the card. This is the amount of in-game "mana" you have to tap into in order to cast the card from your hand. Typically, the casting cost consists of two components, colored and uncolored mana. So the number in the circle is the uncolored mana, more specifically, this means that the mana used to meet this casting cost can be any color. Any symbols other than the number indicate the number of colored mana needed. So, each symbol corresponds to a color (see your basic lands to determine what you can generate) and you'll need to generate one of the specified color for each symbol that's present.
6) Different images on lands are there for atmospherics. They do not impact the function of the cards.