what transaction can help to increase a firm's current ratio?
I'm thinking about payment of accounts payable, and not sure about collection of accounts receivable.....

When you want to know how something will affect a ratio, plug in some numbers and find out.

The problem with the payment of an account payable is that it can increase or decrease the current ratio. It depends on whether the original ratio is under or over 1.0. If you keep in mind that current ratio is all current assets divided by all current liabilities, any reduction or increase of both an asset and a liability by the same amount will do the same thing - it can increase or decrease the ratio, so you can't say definitively that it'll increase it.

As for collecting on a receivable. All that will do is exchange one asset for another and you end up the same place.

Mathematically, in order to increase the answer, you either have to increase the numerator only or decrease the denominator only. (Or increase the numerator by a bigger amount than the denominator, etc.) What type of transaction can do that? I can think of a couple of good quick ones right off the top of my head. Remember that you have five classifications of accounts. Just because they don't affect the equation doesn't mean you can't use the other 3.

purchase of inventory on account
or purchase of temporary investments for cash.....

The purchase of inventory on account is still increasing both a current asset and a current liability by the same amount. The purchase of temporary of investments for cash is still exchanging one current asset for another. In that aspect, they are no different than your original examples.

You seem to want to use the current assets and current liabilities as both parts of the transaction. Just because that is what's in the equation does not mean a transaction involving other types of accounts cannot be involved. As long as one of the accounts fits in the equation, the equation will still change. Can you think of one involving a current asset increase combined with an account that is not a current asset or current liability?