I hope you are familiar with the Verbal aspect since you are asking this question.

1) платить(imperfective) - заплатить( perfective)  = pay, pay for. The idea is compensation. Сегодня я плачу за обед= I am paying for lunch toady) (I receive the meal, the get my money). Они платят мне мало = I am not paid much (They receive my service, but I receive little for it). Ты заплатил за электричество? = Did you pay for the electricity? ( you receive the electricity, they receive the money)

2) оплачивать - оплатить has the connotation of covering the cost of the service. This verb is transitive, it needs a direct object.

Мы должны оплатить этот счёт до конца месяца = We need to pay this bill by the end of the month. In this case, it can't interchange with платить, as there is no idea of compensation, you get the bill, and you cover the cost 

Кто будет оплачивать расходы? =  who will cover the expenses?  

Often you can have a choice of either using платить or оплатить. You can say Кто платит за учёбу? or Кто оплачивает учёбу? In the first sentence, you are saying Who is paying FOR you studies? In the second sentence, you are saying Who pays the bills? 

Notice the use of preposition ЗА

3) уплачивать - уплатить =has connotation of fully covering due payment(often equivalent of money, such as fee, deposit ) on time (that's why the first example that comes to mind is about taxes )

В этом году я уже уплатил все налоги.

Чтобы быть в этом клубе, нужно уплатить взнос.

Florian, actually this question has been asked several times here... So try searching, maybe you will fine more.

платить за...
заплатить за...
я заплатил ему тысячу рублей серебром за эту книгу.

dative - to whom
accusative - sum (1000)
instrumental - means: 'in cash", "in silver", "with a card", "in roubles" (alongside with genitive after sums as above)
за + accusative - for what.

("за-" verbs often have за preposition: заглянул за занавеску)

It is the most common verb. Mostly it describes the physical action: you are giving someone your money for something.
When you say it, I picture you with a purse giving away banknotes and smiling to the seller/waiter etc.
Or you pressing a button an your account decreasing.

It doesn't have other implications. It is universal:
it can be used at place of any other, affecting the picture in my mind with: 1) removing "abstract" context of complex relations 2) introducing that guy with banknotes (more 'humane' and 'physical/mechanical'. Or 'arithmetical', speaking of your purse).

It is also used in figurative meanings, like "they will pay for that!!!!" or "pay with blood":)

No, they are not interchangeable, even though Russians often mix them up.

When you оплачиваешь, there's always some service you receive or a document you pay for. Оплатить проезд/ оплатить ремонт, номер в гостинице. But you can't pay FOR taxes, you pay them off. 

Notice, that Уплатить will be often used to pay the equivalent of money(percents to the bank/ taxes to the government) In  such payments you don't feel gain, but rather a loss 

Thank you for this great answer!

If I understand well, уплатить is sort of "included" in оплатить, and in your last examples, уплатить could be replaced by оплатить:

В этом году я уже оплатил все налоги.

Чтобы быть в этом клубе, нужно оплатить взнос.

Am I right? But maybe it sounds weird