Barcelona or Valencia for one week?
 

We are having trouble deciding which city to visit in June. We are an older couple, not much into museums or lots of churches, but like SAFE neighborhoods where we can walk and have access to public transport. Very much like to be in the area of good mid-range and budget restaurants (usually the places where locals go), interesting shops, and nicer, less expensive places to stop for coffee, wine, cocktails, etc. Like traditional versus new. No car. Will be looking for self-catering or looking for less expensive hotels (less than $100 per night if possible). Beach a plus and would probably do 1-2 day trips out of town (by public transport). We are considering Barcelona and Valencia. Your thoughts?

They are both big cities and I like them both. But it seems to me, you might like the towns on the northern coast more. For example Santander (Cantabria) and Gijón (Asturias) are very nice cities, but smaller and, I dare to say, safer than Barcelona and Valencia.
I don't mean to say that, Barcelona and Valencia are dangerous, but like all the big cities, they do have a drug problem. You will notice it, but I have still felt safe in both of these places.
Santander and Gijón are so much smaller that walking makes sence, especially Gijón. By daily walks, you will learn to know the whole city little by little.
But in short. For a first trip to Spanish east coast, I would prefer Barcelona.

You could esily do both in a week. 4 nights Barcelona, then euromed train to Valencia for three nights. Fly into one and out the other.

I would start in Barcelona if price is the same to have a more relaxing experience in Valencia at the end, as Barcelona is quite busy and not very laid back.

Both places provide a nice vacation, Valencia being more provincial, also spread out but very manageable.

Walksntalks, thanks for the suggestions, but we need to fly back to Arizona directly from spain. Valencia, Barcelona and Madrid are our best choices. Madrid does not seem like a place for us right now, so we are still vacilating between Valencia and Barcelona.

The idea of taking a train from one to the other is interesting and I will research it, but does take a bit away from the relaxation piece of this trip.

More opinions are very much welcome.

For one whole week, I'd probably prefer Barcelona.

As lincasanova pointed out, Valencia is much smaller than Barcelona, so there is less to explore (not talking museums, but simply walking around, different neighborhoods, etc.).
Barcelona is also the hub of a very dense system of public transportation, not only within the city but also with many train connections in all directions (except East :-) ) - so there are a lot of easy day trips possible.

Re. safety, I'd say that you will have to be a cautious of pickpockets in either place - though you will find many that can tell warning stories about Barcelona. Since I never got pickpocketed in more than 20 trips to Spain, I cannot offer any similar stories.