Have been doing the Semana Santa at Sevilla a couple of years ago and it was certainly a memorable experience.
I would suggest you get yourself the official Semana Santa Guide and the newspaper of the day as soon as you arrive. They have all the processions and their itineraries listed. Also get yourself a city map - many the locations in the programme might not be familiar to you.
Then you can select the cofradías you are particularly interested in. Their costume, their notoriety, and their carriages are quite different and some are more famous than others.
The cofradías usually start from their barrio, do the city tour, and return to their barrio, so depending on which one you are particularly interested you should move to the area where they pass. Except for the very city centre where all the cofradías take the same route, you'll find the routes vary for the rest of the tour.
Having said that: the city centre is of course the most crowded area. I found it most moving to go to the place where the procession takes off. There are lesser people, it's the locals who send THEIR cofradia on the tour, and you can watch many moving scenes of families dressing up their men and boys, kissing them farewell, etc. Be there half an hour before the official departure, and you'll have plenty of "Semana Santa for locals".
In general terms, unless you are not well on your feet, I would disadvise to book any of the seated galleries that are also fairly expensive. If you are good at foot, just walk around, mingle with the crowd, walk with the procession when there is a particularly nice part, stay behind when there is something coming up behind you, or change to a different group if you begin to find it boring. If you are not afraid of crowds, there is no problem mingling and find your own way.
Hope you enjoy your trip!
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Jordi