WGT - Trains
If you are flying into Germany then its likely that you will land in Berlin or Frankfurt and then want to take the ICE train down to Leipzig, the trip from Berlin takes about an hour and a quarter whilst the trip from Frankfurt takes a little over three hours. If you are taking the train from the Airport to Leipzig it is advised to book your ticket prior to departure, like flights train tickets are cheaper the earlier you purchase them, expect to pay something like €60-80 for a return trip from Berlin to Leipzig (and sometime 1st Class can be cheaper). If you land and then need to get a ticket then there are machines on the platform with English options.
The Intercontinental Express (ICE) that you want will travel via Berlin Hauptbahnhof or Berlin Südkreuz, if you have time to kill then head to the Hauptbahnhof (there are loads of shops and restaurants and it's only a short distance to the centre of Berlin if you want a walk around). My personal preference is Südkreuz. it's a smallish station (with a couple of food shops) so little chance of getting confused with the size of the Hauptbahnhof (where its been known for platforms to swap from one end to the other). To get to either stations you will need to take the local train from beneath the new airport to either station (when you book your train ticket book from the Airport to Leipzig), check the route map and wait for the short journey into town.
Tickets, there are several option, a flexible ticket will give you the most scope to get to Leipzig without extra cost in Germany (however it will cost more), if you book a non flexible ticket then you have lost your money if you miss your train and you will need to purchase a new one. I would give myself 3 hours from landing to get to Berlin Hauptbahnhof, more if this is your first time / you are nervous of missing your train / connections.
Train platforms in Germany are of course organised, your ticket will likely have a coach number and seat number on it and you will likely see A, B, C etc down the length of the platform, if you check the platform there will be a schedule of trains and a picture of your train and where the coach will be in reference to the A, B, C etc...you need to get yourself to the rough area where your coach will arrive, then as the train pulls into the station get to your coach which will show if your seat is at one end or the other of each coach. You just then need to stow your luggage and find your seat (if someone is sat in your seat then you can ask them to move). The trip back to Berlin is easier because Leipzig is an end of the line stop and it will remain in place for a while after it pulls in whilst the driver changes ends.
Stowing luggage can be fun, no its a pain in arse, if you are lucky to have a luggage store on your coach then it will likely be in the middle of the coach! if the coach is empty you should be ok to stow it there, however on busy trains you might end up standing with your luggage near the doors (esp. if you got on the wrong coach). If you do end up standing dont worry too much the trip is only just over an hour to Leipzig.
It is possible to travel to Leipzig by Train from just about anywhere in Mainland Europe, however again depending on where you are starting will depend on how direct your trip is / how many connections you will have to make. The benefit of travelling by train is you get to take what you can carry and you get to see huge swathes of Europe at the same time, the downside is that you might have to take multiple trains and make many connections which can lead to stress if your train is delayed. It should also be noted that trains in Germany don't run as punctually as you might expect (although still better than trains in the UK).
A number of people from the UK elect to travel to Leipzig by train, travelling from London to Brussels (where many will elect to spend a night) and then on to Cologne, Frankfurt and Leipzig, its about a 9-11 hour journey depending on your route and time you give yourself for connections. From May 2022 there is going to be a sleeper train from London to Berlin which might be an option for some, its then a short trip from Berlin to Leipzig.
Taking into consideration luggage costs, airport transfers, security checks etc its probably a more relaxed journey that travelling by air and you should research all the routes and possibilities before booking (Note: Sometimes 1st Class can be cheaper). You can of course use the trip to explore other parts of Europe, Brussels is a good night out, as are Cologne and various other places between London and Leipzig
Of course taking the train does have a few issues, if you have a large suitcase it can sometimes be a pain to store it (that or you are constantly worrying about it), food and drink are a little restricted and of course you may end up sitting next to somebody you would rather not (this is also true about a plane, however on a plane its only for an hour and a half if flying from the UK)...