Best Neighborhood in Seville?

I honestly wouldn't compare the two cities in that way. Sevilla is larger city than Florence and it gets something like less than half the tourists. So, you really don't need to think about seeking out a laidback vibe or avoiding tourist throngs as much.

Sevilla only really feels touristy in the street immediately adjacent to the Alcazar and Cathedral.

I stayed at a hotel (Soho Boutique Catedral) just north of the cathedral and it was a great location, direct access to the Alcazar and Cathedral, easy to walk north to the Setas, easy to walk across the river to Triana. It would be tempting to compare Triana to the south side of the river in Florence, so you could try that area as well. But, I think you can't go wrong if you're centrally located.

Hello Plectrude! Thanks for the quick response and great info. You brought up some great points. Even through Florence is about half the metro population of Seville, Florence see's WAY more tourists (didn't know that, but it makes sense). Checked out the website of hotel to suggested. Looks like the location is about right in the middle of Cento, Arenal and Santa Cruz neighborhoods. Sounds and looks great! I assume taxi's from the train station to the hotel should be no problem. We'll also do a little more research into the Triana neighborhood. THANKS SO MUCH!

I’ve stayed in Seville’s Barrio Santa Cruz neighborhood which is just north of the Alcazar.
Its maze of narrow medieval streets keeps cars and a lot of pedestrians out because it’s like entering a labyrinth. So, it benefits from being near Seville’s major sights but in a neighborhood most find you can’t drive through and it’s also too disorienting to walk through.

The perfect area to base yourself!

Since you mentioned a taxi to that hotel, I want to note that it's on a pedestrian street so a taxi will drop you about a city block away or so. But, it's an easy ride from the airport and not expensive either.

A few great restaurants and bars I'd like to recommend:

Bodeguita Romero - everything was fantastic. Probably the best tapas experience I've had in Spain.
El Favorito - just a charming tapas bar that isn't crowded on a side street. Great service, good food, good beer, we went a few times.
Lobo López - more of a modern tapas bar, but had some great lobster rolls, great wine, and it has a cool interior design.
braXas - amazing steak and grilled meat.
Uno de Delicias - amazing tortilla

There's also a really nice areas around the "Setas" with lots of outdoor seating and a few really nice places.
Vinoteca Lama La Uva - love this place if you like wine but you'd like a bit of guidance in terms of what to order. Our waiter spoke totally fluent English and was able to help suggest wines.
La Linterna Ciega - great tapas
La Santa - ditto

If you want a neighborhood in Seville that is analogous to Oltrarno in Florence, that would no doubt be Triana. It is right across the iconic Isabel II bridge, and all the tourist sites are within easy walking distance. If you need a reference, go to Google Maps and look at where the Mercado de Triana is in relationship to the Royal Alcazar. The walking distance is 1.3 KM, which to most people would be easy walking distance, and it's a very nice walk if you cross the bridge and go down the riverwalk on the opposite side of the river.

We just got back from Seville, and we stayed in Triana and enjoyed it tremendously. The Mercado de Triana is one of those major sites you would want to see anyway, and the walk across the bridge is very beautiful.

I agree with staying in Triana. It's very atmospheric and within walking distance to the major sites. We stayed in Barrio Santa Cruz but we walked to Triana several times.

Regarding Barrio Santa Cruz, this is a very convenient location, too, but it was very crowded with pedestrians and tour groups. Sometimes we had to stop and wait for the large tour groups to pass us by; other times, we had to barge right through them otherwise we would have been waiting forever. Barrio Santa Cruz is pedestrianized so our taxi dropped us off in a square, and we then walked to our hotel. Not a long walk.

It doesn't get mentioned so much, but I think El Arenal is a good "old town" neighbourhood to consider. It has hotels, so obviously not "non-touristy" entirely, but it doesn't contain any major sights (at least not the really big ones, except perhaps the bullring and along the riverfront) so you won't encounter big crowds of tourists.
But from central Arenal, it is an easy walk to Catedral, Alcazar, Santa Justa in one direction and Alfalfa and museo in others.
Bits will look a bit rough and ready compared to the grander (e.g. Avenida de la Constitución, which does border El Arenal) or prettier (e.g. S Justa) parts of Sevilla. But Arenal is safe and has a huge range of restaurants, bars, etc to visit (although, admitedly, I could write the same about most of central Sevilla).
Anyway, just a suggestion. Also, for any district mentioned, remember thay cover an area and as important as "which barrio" is "which part of which barrio".

walking distance of the major sites in the city.

We stayed at the El Rey Moro and loved the hotel and the location. It definitely fits the standard "walking distance" for most of the highlights, and depending upon your walking fitness, that range can include a lot of really lovely stuff to see.

I'd also suggest a place we loved to eat at called La Brunilda for tapas. Slightly off the beaten track, but an easy walk from the Cathedral area and popular enough we made a reservation the second time we went there after they squeezed us in the first time.