How do Shia Muslims pray when travelling?
On a long enough trip, Dhuhr, Asr and Isha are prayed as 2 rak'ahs instead of 4. This is called qasr. Fajr and Maghrib stay the same.
In Ja'fari practice, qasr is not a choice. If your trip counts, you pray 2.
When does a trip count?
Distance. The trip is about 44 km (8 farsakh) or more one way. A shorter trip also counts if you go at least about 22 km and come back, so the whole trip is 44 km or more. Count from the edge of your town, not from your door.
You plan it from the start. You set out meaning to go that far.
You stay less than 10 days. If you plan to stay in one place for 10 days or more, you pray in full there.
You are away from home. Back in your home town, you pray in full again.
There are a few other rules, such as for people whose job is travelling, like drivers. Ask your marja' about these.
What about fasting?
If a trip shortens your prayer, you also do not fast that day in Ramadan. You make the day up later.
The times don't change
Travel changes how long the prayer is, not when it is. Use the prayer times for the city you are in, not the one you left.
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Next guide: why Shia break the fast later →
Sources: Office of Ayatollah al-Sistani, Islamic Laws, on the prayer of a traveller. Other marja' may give the distance a little differently.