How to use public transport inside European cities (short visits)?

If you can tell where you're planning to go it will be easier to tell but I can give you some tips for both Barcelona and Madrid in case you're coming to Spain.

In both cities I believe that the tourist ticket is too expensive because most of the places that everyone wants to visit are within walking distance from one another. I would recommend you the same in both cities, buying a 10 trips ticket.

In Barcelona (1 zone which covers the city) it costs 9.90 € and you can use it for the metro, bus and some train services around the whole city, also you can change transportations within 30 minutes from the activation without needing a new ticket. It's much cheaper than buying single tickets because they are 2.15€ each.

In Madrid is similar but it costs 12.30€ (1.50-2 € for a single ticket) and you can't change between metro and bus without using a new ticket.

I would recommend you to check if something like this exists in the cities that you're planning to visit because, unless you're going to use public transport a lot, it's much cheaper than tourist passes.

The answer is a big it depends.

But to give you a bit of an overview, you will find a mix between:

• Traditional paper tickets, still in use in many places and often good enough, in Paris for example. Even for a week and if I am staying in the centre, I would stick to the T+ ticket (you can buy 10 of those at once at a discount). As per Adolfo's answer, Madrid and Barcelona are similar.
• Simple day passes. In some places (Berlin), it's very good value, you are saving money starting with the third trip of the day. Even if you are only staying one day, there is no reason to buy single-journey tickets there. In others (Amsterdam), you have to travel a bit more to save money but it can still be worth it depending on how far/often you travel and of course it's very convenient. In both cities, there are also multi-day variants that are also very cheap.
• Tourist-oriented multi-day passes, aggressively marketed with some discounts on famous sights and the like. Convenient but usually not the cheapest solution if you are only interested in public transports.
• Pay-as-you go systems like the Oyster card in London or the OV-chipkaart in the Netherlands. Usually good value even if there are some sunk costs and the details matter. London was actually among the first cities (I know of) to introduce something like that but it is already moving on to
• Contactless payment directly with your credit/debit card. Hard to beat for convenience but with a foreign card I would be concerned about compatibility and exchange rate/fees. For this reason, I haven't used this in public transports yet.
• Free public transport. That's a thing now even if the examples I know are not necessarily huge destinations. One that you might visit is Talinn in Estonia but it's only free for residents, not for visitors.

I assume you will mostly travel overland but do beware that airport transfers are usually more complicated/expensive and often fall under different rules.

Your question is very wide and will likely be closed for that reason, unless you can re-write it to fit into the rules of this site.

A lot depends on which cities you visit and where in the cities you stay and how much you want to move around in public transport once there.

It is very well possible to find a hotel or hostel near your arrival railway station, use a tour to see the city and do the rest of your 'traveling' walking, even in a city as big as Paris and London. On the other hand, if you stay on the edge of a smaller city and all you want to do is spread out over the city or even the countryside around it, you may need a transport pass or a day pass for transport in a smaller city.

And there is no 'standard' pass or card for cities in Europe. But when you look ahead of time you might find day passes or multi day passes, often with other discounts included, in most of the cities many tourists visit. But compare with single rides, single rides on local passes and sometimes available 'ten ride' tickets (which can have any number of rides.)

If you love walking, going sight seeing in cities on foot can be very satisfying if time consuming. On the other hand, if you are very short on time and have enough money, you might chose the city tours on 'one day' or '24 hour' passes and not need local transport at all.