Go with wood and nice rug.

Wood all the way!

Depends on where you live...here in Western NY, I recommend tile that blends nicely with the rest of the wood for a seamless transition.

I prefer wood.

It depends what style you are going for .. If its traditional then wood if its modern then tile .. I think wood is overrated , since everyone has wood flooring . Check some tile ideas .. Again depends on the look you are going for

Your space will look seamless with a single application like wood.

Thank you so much for all the advice! We live in Central California.
It's this tile or wood. We would not replace the tile with a different tile.
My question with doing wood throughout...then you have an entry rug, a dining rug, and a living room rug?

Wood. I personally wouldn't put a rug in the entry way as it will get dirty with people stepping on it getting in and out of the house. I would only use a rug if you feel that you need to ground the space or don't like your feet against the cold floors. I don't think it is mandatory though just because you have the same flooring throughout.

There are also some great tile options that look like wood. Very durable and beautiful.

Wood through out!

You could have a rug in all three. The only room I think you really need one for sure is the living room. Other 2 rooms optional.

Here are some photos of open concepts

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For your entrance you need durability. Wood will look very nice, however you can achieve the look and maintain the durability by using a wood floor pattern tile, 4" width x min 12" length.

Wood.

definitely wood, otherwise it won't flow visually I did wood floors throughout the first floor.
I think you will noticed how large the foyer will now look once you have a wood floor, suddenly it won't feel like the foyer stops at the tile.

Wood for sure

Tile! Wood at the front door bad idea asking for scratches and dents from the rocks in your shoes etc