How to book/organise train travel in Germany and Belgium

Tickets for almost all your listed trips can be bought with the Deutsche Bahn (German Railways). Buying them well in advance you can benefit from the availability of heavily discounted Sparpreise (Saver tickets). Easy and reliable process: payment by credit card and an e-mailed ticket for printing at home.
Buy Brugge to Liège at the railway station. Domestic trains within Belgium don't come with advance discounts.

Added:
In the Flemish part of Belgium Liège is known as »Luik«.

RailEurope doesn't operate any trains. They sell overpriced train tickets, and from what I've read their customer service are clueless about the reality of train travel.
You should try to book directly with the operators: Deutsche Bahn in Germany and SNCB/NMBS in Belgium. As a rule of thumb, it's best to book with the national operator for the country where your journey starts.
If that's not an option for some reason, you can try trainline.eu and loco2.com.

You can save a lot of money on all the legs mentioned above I've you book ahead and get some of the DB Sparpreis or Supersparpreis tickets. The only exception is the Brügge - Liege leg: The Belgian Railways doesn't sell discounted advance and non-flex tickets, but instead all domestic SNCB/NMBS train tickets in are inexpensive.

Hello Pip

Are either or both of you over 65? If so the travel in Belgium gets very inexpensive.

Lucky you going to Spa for the best race in the calendar. I hope you get a great viewpoint and that the weather gods cooperate.

If you look around here you won't find much love for RailEurope either, despite our host having a relationship with them.

Entirely Belgian journeys, Brugge to Liege for example, don't vary in price and are inexpensive and straightforward. They don't include buses or trams. You will be crossing the language divide between Flemish and French, and between the two halves of the country, Flanders to Wallonia. Throughout Flanders the same bus and tram company, de Lijn, operates in almost all the region, so a day ticket issued in Brugge/Bruges is also valid in Ostend, Gent and Antwerpen - but not anywhere in Wallonia or Brussels.

In Liege you will need to use TEC for local transportation.

The trip from Liege to Berlin should be booked ahead from the SNCB (Belgian) or DBahn (German) railways. Use the hints in one of the pages here to plan the train travel and how to use the DBahn:- https://www.ricksteves.com/travel-tips/transportation/trains/online-schedules

The same applies for Berlin - Frankfurt am Main.

Local (Regional) options are easier Frankfurt (airport or city??) to Trier, and from Koblenz to Brugge/Bruges.

Are you driving or taking a cruise from Trier to Koblenz?

I see that ticket prices are already elevated for the Monday after the Belgian Grand Prix, Monday Sept 2, the 7:12 departure from Liege to Berlin is 139.80 EUR for 2. Wait until 9:12 and it is 89.80 for 2, and wait until Tuesday Sept 3 and they are 79.80 for 2. These are advance purchase nonrefundable (except with a fee) bought now.

Frankfurt airport to Trier can be bought on arrival for 52 EUR with a "Quer durch Lands" ticket for 2. You must use regional trains only for this trip, but because of the rail network, you do not save time using fast national trains.

Berlin to Frankfurt can be done for as low as 51.80 EUR on a direct ICE, but it depends on travel date. That is the price for Thursday Sept 5, but on Friday, the price goes up, with the cheapest for 95.80 and most departures much higher. On Saturday a little lower, etc.

Koblenz to Brussels can be done for 59.80 for 2.

Curious how you go from Trier to Koblenz. Bicycle? River boat? It can be done completely flexibly by rail with a Rheinland-Pfalz ticket for 30 EUR for 2, traveling after 9 am on week days.

I think you can see that if you can decide your schedule now, you can save quite a bit of money over an expensive Eurail Global Pass. If you do decide on that, you only need a 3-day pass which is 492 USD, that will cover the major legs of Koblenz to Brugge, Liege to Berlin, and Berlin to Frankfurt. With advance tickets, you could do the same trips for the equivalent of 230 USD, less than half, but at the cost of being locked in to a schedule.

If you want the flexibility of a pass, the German Rail Pass would be a better deal than Eurail. It is valid on the ICE trains between Cologne and Brussels and Liege to Cologne, as well as all over Germany of course. a 3-day Twin Flex Pass 2nd class is 292 EUR, or 4 days for 314 EUR. As has been discussed above, train tickets for inside Belgium are not expensive, and trains very frequent.
https://www.bahn.com/en/view/offers/passes/german-rail-pass-conditions.shtml?dbkanal_007=L04_S02_D002_KIN0060_GERMAN-RAIL-KONDITIONEN_LZ01

It does not matter how much you will spend on trains because you plan to visit Berlin in one of the most expensive weeks (IFA fair, festival, congresses). Based on accomodation costs the train costs will not matter.

Tip if you want to save money: do Berlin before first week of September.

thanks all for great advice :)