All are a nightmare-though of course railfans will contest this.

Its STAFF shortage-which makes any trip via most airports (and yes, even here in DE=Germany) a nightmare-but maybe its a little better in autumn (less tourists, perhaps more staff working)

It depend what you call cheap-but HAM is not overly blessed with very cheap flites to these places-there are, but mostly to islands in the sunny mediterranean. Check RYANair and EAsyjet what they can offer-but both are also partly in strike or forced to reduce timings due to shortages. EZ is doable, Ryan is a disgrace of an airline-but very cheap-IF you can manage their surcharges. Also search for other nearby airprots-they like to use the budget ones far away.

For better standard-and not that cheap-there is EuroWings, in fact local of LH=Lufthansa, STAR alliance, so you can even get the miles for AC=AirCan.

But -yes, I am in DE now too for a few days- DB trains is also a disgrace. NO train ever on time and also many cancellations. Now in summer local trains are hugely overcrowded by an extreme cheap ticket (9€ for the whole month), but thats over by then.

By train in such a short time: Pra, Ams very doable, Par moreorless-but cost anarm&aleg, Nap too far away. You did not tell if you want to go out+come back every time or not-but these places are too far apart to easy combine them (except Par-Ams). Simply check on bahn.com what it looks like and what they want in price-maybe its not yet on-then use a date in sept as rough guide.

However-all in all it will grossly also depend on the covid situation by then-DE is about the only country in mid/north EUR that still wants you to wear masks in transit and anyone says in autumn with colder weather it will get worse again. There may even be much more strict regulations and travel bans by then.