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: 
I was thinking, if they are different life beings and have different life standards and materials use, what they have in common? Because one of the civilizations could valerate gold, whereas the others simply don't care.
But what they have in common? 
Both have difficulties and cost to travel between their two planets. 
The exchange rate could be a trip from one planet to the other. **They can calculate the equivalent in 1kg per trip.** 

I assume that the cost of travel is more or less the same in each civilization while being a little cheaper for the most advanced one. But they are more advanced, is logical they have a more valuable exchange. 

For example:

A car cost 50,000€, and the European Union can put 1 kg in orbit for 9047€ (according to Wikipedia, Ariane 6 cargo capacity and approx launch cost are 10500 kg and € 95 million). The car cost (50000/9047 = 5.52) 5.52kg per trip.

Meanwhile, in Russia, they are selling a laptop, about 40000 Russian Rubles. The Soyuz have a charge capacity of 7200kg and a cost of 3200 million Rubles. 1 kg per trip cost 444444 Rubles. The laptop cost 0.09 kg per trip.

Those calculations are approximate and the capacity and the cost of the launches and capacity of the rockets are also approximate, the source is Wikipedia.

If we convert the currency (the laptop cost approximately 600€, 40000 Rubles), the car is 83 times more valuable than the laptop. So, 0.09 kg per trip 83 times more, are 7,5 kg per trip. The Europeans have they car for 5,2 kg per trip. 
**The exchange rate is 1.44 Russian kg per trip for 1 European kg per trip.** I think it's a good ratio.
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.