と and や are used to connect two or more nouns.
Most of the time, と can only be used for a fixed number of items like:
キーボードとマウスがいる。
We need a keyboard and a mouse.
But や is used when there is a variable/unknown length like:
キーボードやマウス、それにLANケーブルとかもいるかも。
We need a keyboard, a mouse, and probably LAN cables too.
や can also have the sense of "OR" or "AND/OR" as in:
日本で寿司や刺身を食べたことがまだありません。
I haven't tried sushi and/or sashimi in Japan yet.
But と only means "AND":
日本で寿司と刺身を食べたことがまだありません。
I haven't tried sushi and sashimi in Japan yet.
そして is used as a conjunction between two phrases and has the sense of "then ..."
~たり is used to connect two verbs and is used in the sense of giving examples, like Lukman mentioned in the comments:
泣いたり笑ったりするのは生きてる証拠さ。
~くて is used to connect two adjectives like:
彼女は美しくて、格好いい。
She is pretty and has wonderful style.
And yes, ~し~し is used to connect two adjectives, with exclusive positive or negative senses:
この部屋は綺麗だし広いし間取りもいいから人気がある。
This room is clean, spacious, and also well partitioned, so most of the people like it.
あの部屋は汚いし暗いし駅からも遠いからなかなか借り手が見つかりません。
That room is (a bit) dirty, (kind of) dark, and far from the station, so nobody bothers to rent it.
~も is normally used in the sense of "too / also", but sometimes it is used in the sense of "AND":
これもそれもあれも全部欲しいな。
I'd like to have this, that, and also that one, basically everything.
And there are some more forms like ~かつ~ ("plus ..." or "not only ... but also ..."):
必要かつ十分
necessary and sufficient
Also, 及び, 並びに, ともに, and おまけに have a similar usage to "AND".