Question: There are two worlds of more or less equivalent technological level, say a difference of only a few decades and no more than 50 years. A first contact event occurs and after the hoopla dies down trading starts. 

Initially, barter is the primary form of trade but this quickly becomes cumbersome.

How would the two economies first establish a currency exchange and exchange rate?

Some commodities are similar and occur in similar quantities, particularly minerals gold, iron etc. Other commodities are unique to one civilization or the other; plants and animals unique to each planet, unique tech developed by one and not the other.

Some restrictions:

* Both planets have one currency or at least a dominant currency (like the USD)
* Neither planet is in a "dominant" economic or technological position.
* Travel between the planets takes a few days
* There is a moderate level of trust in business dealings but it is not clear how laws apply to off worlders on each planet.

Ideas I've toyed with:

* comparable commodity-based exchange, ie. gold standard type exchanges, but this seems archaic for technologically advanced worlds and would anyone really want to transport heavy gold out of the gravity well of a planet
* some type of crypto type exchange, but this seems problematic with distance and not very conducive to trade as the price fluctuates

Answer: 
As bitcoin history shows, currency exchanges will emerge very quickly as soon as the currency is considered to have some nonzero value, despite exchange rate fluctuations. The rates on all these exchanges tend to quickly converge, and are used as a basis even for other people who trade directly. Even after failures of large exchanges like MtGox, the trade went on.

Unlike Bitcoin, you have equivalent goods on both planets (gold, drinking water, rocket fuel, ...), so trader can use them to calculate the estimated value of foreign currency. 

When the interplanetary trade gets non-negligible and it gets regulated, these exchanges gradually enter the regulated regime and precautions to stabilize the exchange rate are made. I assume here both governemnts view the interplanetary trade as something good or neutral, not try to stop it.
During the barter phase, when you were trading a ship full of grain for some crates of microprocessors, you established the relative values of the commodities. Now that you have this information you can start trading with tokens, like the federal reserve notes. The parties will have to establish a clearing house for transactions, the parities between the tokens will have to be agreed upon, either by state decrees or by letting the tokens' value float in the markets or by having a common currency.

Until you a have an agreed-upon clearing house you will still need to resort to barter.