Train or rental car in Great Britain - which is better?

Which is better depends entirely on what you want to see. You may do best with a combination. For instance, you don't need or want a car in London or Edinburgh (or indeed other cities), but for the Cotswolds, it makes it much easier.

A bigger concern is that you're trying to see both Great Britain and Ireland in a short time. If you really want to do this, book a multi city ticket, into London and out of Dublin or Shannon (or vice versa). Otherwise, you lose a day getting back to London. However, given how much there is to see in each, you won't have trouble filling 13 days in just England and Scotland. And if you do want to do that, look at flying home from Edinburgh or Glasgow instead of London, for the same reason - avoids backtracking merely to spend a night in an airport hotel.

If you stay in London for a day or two you will not need a car. If you plan on heading North from there, I would suggest the train to York (on your way to Scotland) and then you could rent a car from there. The Yorkshire Moors, Dales or even the Lake District are wonderful and you would be better off with a car at this point to visit the villages. From there you could head up to Scotland by car. Taking a car across to Ireland is more complicated. You would be better to do a budget flight to Ireland and rent a car from there. IMO 13 days will go by fast just seeing sights in England and Scotland. I would save one country for another trip. Either do Ireland for a week then budget airline to England or do just England and Scotland.

If you only have 13 days, you don't have time to see much if any rural sights or small towns. Thats just enough to hit 3 maybe 4 cities. It will take most of a day in transit from one place to another as it is, driving or rail.

If you only have 13 days, and your first and last days are half-days because of your flight over and flight back home, that leaves you 11 days to split between England, Scotland, and Ireland. That's 4 days for England, 4 days for Scotland, and 3 days for Ireland.
(Or change around the days; 3 days for England, 3 days for Scotland, 5 days for Ireland)

You cannot get the true flavor of any of those places in that period of time.

In May 2016, my husband and I spent a little over three weeks in England, divided between London and various locations. We still did not scratch the surface of what is available to see in England.

I suggest, IMHO, you may want to consider taking one location off the table (Ireland) and save it for another trip. If it were me, I would take 2 locations off the table.

You will not be in England long enough to need a car. You will probably see some sights that are best reached by train anyhow. (Maybe Windsor Castle, Hampton Court Palace, for example.) You probably want to do London (3 days), train to York (1 night, 1 day), train to Edinburgh (2 days), train to Glasgow (1 day), with maybe a (one-day) stop in between. You would not want to do this by car, unless you had two more weeks to add to your vacation. In that case, you would have the time to explore out-of-the-way places.

Fly to Ireland from Glasgow airport. Fly home from Ireland, maybe.

During your time in London, do not rent a car. Do day trips out of town for those things you want to see, on the train, or by private tour companies.

Tour companies for day trips out of London:
The London Walks Company www.walks.com
Golden Tours/Gray Line: www.goldentours.com
Evan Evans Tours: www.EvanEvansTours.com

If you can give us a detailed list of the places/things you are planning to see, we can get a better sense of whether you will need a car or not. I already doubt that you do. You can travel faster by train in most cases.

You wrote, "get a rental car? I am thinking that a lot of the sights are kind of "rural" or out of the way, so the public transportation might not be feasible on the island?"

This is not necessarily true. It depends upon which sights are on your list.

England and Scotland are somewhat different in this respect.

Many of England's sights are easily reached by train lines. Even castles that are a good distance outside of London are served by rail. Or rail plus very short taxi ride or walk.

You could do your England portion of your trip all by train. Then after taking the train to Edinburgh, and spending a couple of days in Edinburgh, pick up a car there to explore the Scottish countryside for a couple of days.

What are the sights you are wanting to see in England?
What are the sights you are wanting to see in Scotland?

Wow, thanks for all the replies. We just started looking into the "most important" sights in Great Britain, and I have to admit, there is A LOT! So here is a bit more info on our travel idea.
We usually do AirBnB for our overnights. Many of these require at least 2 nights in a row, which is fine with is, we don't want to pack up and go to a new place for every single night.
If someone could advise on what to see in England (London for 2 days obviously), then traveling up to Scotland. I guess we can forget about Ireland. What would be the bucket list or something you recommend? We are easy travelers, like to do LOTS of walking, are interested in nature, castles, art (me) and beer (hubby). We just spent 2 weeks in Italy, so art can take a back seat this time... We generally like to just walk around a city, enjoy a cafe and do people watching.
And we just love the spectacular ruggedness and the old castles in Scotland...
Thanks for all the recommendations so far!

For ruined Scottish Castles I would recommend Tantallon and Dirleton, Cragmillar outside of Edinburgh, and Dunnottar.

You need the Rick Steves Great Britain guidebook, now on sale if you buy here on Rick's website:
https://store.ricksteves.com/shop/p/britain-guidebook
Now on sale, 20% off through 11/8.

You need to do some research on your own, using this book. The guidebook is an easy and fun read with good maps, and suggestions for hotels/B&B's and restaurants/pubs.

Use it to make your own personal list of "must-sees".

We can make suggestions until we are blue in the face, but in the end, the choices are yours to make.

I think you would get a huge benefit from reading Rick's guidebook.

Following on Rebecca's suggestion, don't just get Rick's book. Look at as many as possible (raid your library). And look at travel videos, Youtube, etc. Cast a wide net, and see what draws you.

Rick's videos are here: https://www.ricksteves.com/watch-read-listen/video/tv-show

And if you're interested in some less trodden places, my trip report on Glasgow, Manchester, Liverpool, and Chester is here: https://community.ricksteves.com/travel-forum/trip-reports/uk-trip-report-glasgow-manchester-liverpool-in-september-2016

Only when you've decided what you want to see can you determine transportation. Similarly, don't book flights until you know your itinerary.

Great advice from Harold! Do watch the Rick Steves videos he suggested. And do read his excellent trip report.

Wow, so many replies! I actually found Rick Steve's book on GB (we had the Europe book, which only included London), so now I can read up and follow all your guys' links. We do have some time, so I will report back what we found with your help and the books/reports/videos. Thank you so much!