What is the difference between 租, 租用, 租出, and 租赁, all meaning "to rent"?
Does it matter if you are the person who rents or the person is renting

Yes, there are differences. The main verb is always 租, with different meaning based on different directional complements:

租出去 = to rent out, to lease

You use this term when the subject is the one who lends something for a fee. Example:

我想把我的房子租出去 I want to rent my house (to somebody else)


租下来 = to take out a rent

You use this term when the subject is the one who borrows something for a fee. Example:

这套房子离你公司很近，还是赶快租下来吧 This house is very close to your workplace, let's rent it now (let's take it)


Other words you mention:

租赁 = (to) rent, (to) lease —— dictionaries list this as a verb similar to 租用 (to take on lease) and 出租 (to give on lease), though in practice, it's often used in noun phrases such as 租赁合同 lease contract, 租赁条件 lease conditions, etc.

租用 = to rent —— with the result complement 用，rent and use，I think this is more common in Taiwan