が知っています？ or　を知っています？
 

Is が 知っています or　を 知っています correct?

I've been told that when talking of an ability and potential verbs I should use が. Like が　読めます。

But in regards to 知っています and　分かります I get confused, due to the similar meanings.

Can somebody please explain whether it is が or を that precedes both 知っています and　分かります?

It goes:
~を知っている
~が分かる

Thank you! :)

Exceptions:

1. To answer the question "Who knows (something)?", one must always say:

「～～が知っています。」

In fact, that question itself is also asked with が.

2. Inside a relative clause:

「私が知っている人が来月アフリカに行くんですよ。」

The が here can be replaced by の, but never by は.

So this means that it is always を 知っている but only when its preceded by だれ　which normally follows on with　が anyway it appears to take the form of が知っています?
And when answering a question with が you would have to use が and not は in the answer, right?

Can を be used in a relative clause at all?
For example 私を買ったかばんは～～
Or would it just be 私が買ったかばん・私の買ったかばん?

That is correct.
If you wrote out the full sentence it would be (Someone)が(Something)を知っています.
But because you don't need to say (Something)を if it is already known it becomes (Someone)が知っています.
知る is a transitive verb if that helps you.

You can use を in relative clauses.
A relative clause is basically just a sentence within a sentence.
For example in 両親を殺した人間を見つける (“who killed”の検索結果（77 件）：英辞郎 on the WEB：スペースアルク)
両親を殺した is modifying 人間.

However you cannot use it as you have done in your example above... That is unless you lived in a world where a bag could buy you.
買う is another transitive verb, which means it should take a direct object with the を particle.
(something)を買う = To buy (something)

私が買ったかばん and 私の買ったかばん are the correct versions.

That's a great explanation, I think I've understood it now. Thank you =D

Just curious if these sentences are correct?

Q. 誰があのパン屋の名前を知っているんですか？

A1. 僕が知っているんです。

A2. 僕があのパン屋の名前知っているんです。

I do not understand the confusion as to が and を, and what this has to do with relative clauses: が marks the noun that is performing the verb action ("subject"), を the noun the verb action is being done to ("object"). That is, when used with 知る, が marks the person who knows something and を the fact this person knows.
Whether in relative sentences or not, 私を知るwhich obviously means "someone knows 'I'", which is not what we want to say. And it is a general rule that 誰 cannot be used with は, but needs が. This has nothing to do with 知る.
Similarly, 私を分かる doesn't mean "I understand" either. 分かる could be said to be an "exception" (as for が), but as for 知る and many other verbs, they behave as expected with が and を.

Your sentences seem to be alright.

Thanks, blutorange.