Summer 2016 trip to Ireland and Great Britain itinerary and questions

One obvious wrinkle in your proposed itinerary is the requirement to fly in and out of Dublin. If you're not already committed to that you'd be better off doing your trip as an open jaw: flying into Dublin and returning from London, thus saving yourself a day's travel returning to your point of origin.
Other than that your plan looks pretty sound to me.

I think your itinerary looks good. However, I agree that you should look into flying home from London so you don't have to waste time and money getting back to Dublin. I also see no reason to allocate another two nights to Dublin, which I did not like all that much. I would allocate them to London, which has so many great things to see. Or you could use them for day trips from London. Another option would be to use the two nights in Dingle while you are in Ireland.

If you have to fly home from Dublin, consider options of flying from UK to Dublin.

A great site for transportation planning is Rome2Rio - great resource to get ideas of transportation options: http://www.rome2rio.com/

In Edinburgh I used Rabbies tours to take a Highland tour and was very pleased with the service: https://rabbies.com/

Some budget hotel ideas would be to search Ibis or MotelOne. MotelOne is my favorite, but Ibis can have some very good budget options in many locations.
http://www.motel-one.com/en/
http://www.ibis.com/gb/booking/hotels-list.shtml

It looks like a really good plan, although I can't comment on visiting Ireland using public transportation.

From Dublin, you might add a day trip to Glendalough; the monastic complex is beautiful, as is the setting, and it will add something that I don't see in your other stops. Also, if you have to return to Dublin to fly home, you might want to investigate some of the low-cost airlines, either returning to London to go to Dublin or flying from an airport closer to Bath. Rome2Rio and skyscanner.com are good sources for info.

Finally, to answer your question about booking lodging ahead of time, I recommend it since you mention wanting to keep costs low. B&Bs will be cheaper than hotels. You can find lodging at the last minute, but the high quality, inexpensive places are typically not available for a last-minute arrivals. Also, I wouldn't want to spend touring time looking for a place to sleep, particularly if I'm relying on public transportation to hunt for a room.

Thanks for all the advice. I did look into an open jaws flight, but the ones I found are much more expensive than the round trip into/out of Dublin. I am investigating flying from London back to Dublin. One option is to fly from London to Dublin on a cheap one-way ticket, have a 2 hour layover, then fly Dublin to SFO. Does anyone know if we would have to go through customs in both London and Dublin, and if so, if a 2 hour layover in Dublin is sufficient? Thanks again for your help!

You would go through a passport check changing in Dublin as there is no general airside transit facility. However, some of the time this is via a separate corridor to arrivals.

"I did look into an open jaws flight, but the ones I found are much more expensive than the round trip into/out of Dublin."

You may already know this (in which case disregard it), but you should be looking at "multi-city" flights, not two one-way flights. And even if it is more expensive, your time is also worth something, as is the comfort of having all the flights on one ticket.

"One option is to fly from London to Dublin on a cheap one-way ticket, have a 2 hour layover, then fly Dublin to SFO."

How lucky do you feel? If there's any disruption in your plan, you will miss your Dublin to SFO flight. You will then have to buy a new, one way transatlantic ticket at last minute prices. There's no way I'd try to make this on a same-day connection with two separate tickets. If you are going to do this, fly in the night before. I had to do this going from Sicily to Rome on one ticket, then Rome to the US on another ticket; the peace of mind was well worth losing a night in Sicily.

Also, most (although not all) Dublin to US flights have "preclearance," meaning that you go through US immigration and customs in Dublin; your arrival in the US is then like a domestic arrival. So, you need sufficient time in DUB for this.