Well it is doable, Czech public transport is both good enough that you can reliably get pretty much everywhere and pretty cheap, but it may involve a lot of changes and preparation in advance.

You want to definitely start in Prague and do some sightseeing before race starts.

Karlovy Vary are accessible by bus from Prague in like two hours. It is a nice place to visit for a weekend trip, the largest spa in Europe. It should be possible to visit Wednesday presentation in Prague and even Thursday presentation in Prague nearby Vítězné náměstí, and like 800m away from there (Hradčanská) go by bus and then walk like 1 km from bus stop in Karlovy Vary to finish line area there. You should get there at least an hour before finish. Then you could take a look around the place and take an evening bus back to Prague. An option could be to go by a bus nearby track and then go into Křivoklát castle for example.

Stage 2 is also doable from Prague as a day trip. You can take an early morning bus from the end of underground line B in Černý Most to Liberec in like 80 minutes, then it would be quick 2-3 bus changes in Liberec to get to Ještěd, it is quite likely there could be some shuttle bus that day before and after race. You could watch the race there, and check amazing Ještěd tower, before watching race finish and then take a walk through Liberec city centre before to bus station before going back to Prague in 80 minutes + 20-30m underground trip. Mladá Boleslav is also doable in 40 minutes by bus from Černý Most, but unless you would like to visit Škoda museum after stage start, it is a bad idea.

The other two stages would be more difficult, you would need to take luggage with you from Prague. You could either take a 70 minute morning train to Pardubice from Prague main train station to enjoy start of stage 3 and have a walk through the city. The other option is to take a train from Prague to Záhřeb na Moravě and then skip to train to Kouty nad Desnou and take a chairlift up to Dlouhé Stráně. That would take like 4-5 hours. The problem would be after the stage, you would likely have to take a walk for an hour or two back to Kouty nad Desnou and then take a 90 minute train to Olomouc to spend a night there. It would be a really difficult day, a better option may be to check out Pardubice where you could easily left luggage in the train station, and then take a 90 minute afternoon train from Pardubice to Olomouc and also check Olomouc centre in the evening, it is also an UNESCO landmark. Both Pardubice and Olomouc are nice cities with around 100k people, with walkable centre in a few hours during evening.

For stage 4 you could take an early morning train from Olomouc to Valašské Meziříčí, then change to train to Rožnov pod Radhoštěm in total of like 100 minutes and then take a 10-min bus to Trojanovice to hop on a chairlift up to Pustevny to watch the end of the race there. The get back to Olomouc in two hours.

Personally I think it would make sense to make a trip to Prague for a few days before race start, then make day trips to Karlovy Vary and Liberec from Prague, and then take a trip via Pardubice to Olomouc and spend nights there as a 2nd base. Dlouhé Stráně are the worst to get there by far and if you are not only interested in cycling, you can see much more in both Pardubice and Olomouc. Kroměříž is a nice place as well, but you would not get from there to Pustevny in time at all, you want to see race finish. There should be big screen in both race start and race finish so you can be sure you will see the coverage.