Spending One Month in Great Britain From Southampton

First of all, there are ATMs everywhere! Don't even think twice about that. They make it very easy for your to withdraw your $$$ in pounds and distribute it to the British economy!
I am also a walker/hiker and can make a few suggestions...
If you really want to do a lot of walking and hiking, consider adding time to the Lake District. You have two full days currently scheduled and, since it rains there frequently, I would give that stop one or two more days just in case the weather doesn't cooperate. It's my favorite place in the world to hike, and I truly mean that! There are many short hikes and long hikes, strenuous and meandering. One of my personal favorites is the hike to the top of Haystacks, near Buttermere. You can easily get there by the local bus service. It's rather strenuous but if you take it slow, as I do, the views from the top are unmatched. I also enjoy the hike up Skiddaw or Blencathra. Hiking along Derwentwater on the east side going south from Keswick is also really nice. Don't miss the Castlerill Stone Circle just east of town. From there you can hike to Walla Crag and Ashness Bridge...a lovely hike with great views over Derwentwater Walla Crag Hike I know that the Rick Steves route is up the Catbells, and although it is a nice hike, it's also very overrun with hoards of people. I prefer the more remote locations.
There is also a lot of good walking in Cornwall. The coast path wraps all the way around the peninsula and you can easily do small, very scenic pieces of it. This link has lots of great ideas...Southwest Coast Path.
We have also done a bit of walking on the Yorkshire Moors - windswept and lovely! Yorkshire Moors.
I have heard that Scotland is amazing but have never been able to work that into our plans yet. It might be worth checking out!
Enjoy your trip!

I agree with both previous posts, particularly the idea of a few pints to get you started!
As you are arriving in Southampton a short ferry ride to the Isle Of Wight would be an ideal place to start. It is good for walking, plenty of history (Osborne House, Carisbrooke Castle) and a Botanic Garden at Ventnor. In addition the climate is very benign, not Florida but certainly a lot milder than much of Britain. If you are in Britain before 18th May the Isle of Wight Walking Festival will be running but there are lots of marked footpaths if you arrive later. If you do decide to visit the Island Mimosa Lodge in Cowes is extremely comfortable and with a very friendly welcome.

Why not drive and have the flexibility? Coming up to Keswick is a great idea. I live about 45 mins away from there. If you are coming up to Keswick why not go the whole hog and get yourself up to Scotland ? The west coast of Scotland is the most rugged wild place. I was staying in a lovely castle hotel last year and gave some advice to some Americans and they said it made their holiday. My advice was to go to inverpolly...go to a place called reeth and then to stoer light house...

My advice to any visitor to the UK is to drive it is so much more flexible public transport isn't brilliant here and me personally I would never travel anywhere in the UK by public transport..it just can't be relied on.

I would also say that a month is a long time you could really do a lot. What are you interested in, history? Art? Shopping? Don't worry about ATMs, we might have third world public transport but we do have the worlds supply of ATMs...

Thank to all for your helpful advice!

As far as driving, I heard the expense and the hassle of parking outweighed the convenience of the car. Also, as my wife has to get back to work after a week and I don't, so I was planning on backpacking north to Scotland over the remaining three weeks. Any thoughts on that adventure idea? How safe is it for a single, strapping ex-military guy like me (LOL) to take off for the Scottish hinterlands with a backpack and a map?

I wouldn't say parking in the UK is expensive...wow have you been to the states! Most ordinary parking is free. Only in big cities would you pay for parking or perhaps a tourist attraction. Also it's not massively expensive and generally if your staying at a hotel here in the UK (unless it's LOndon) generally parking is free again unlike the states!

Backpacking around Scotland, great idea. But Scotland is remote, shops and towns in the highlands few and far between in some areas. People are reluctant to pick up hitch hikers especially lone males it's not really a culture anymore of hitch hiking. I'm ex military ex police and I would still advise a car. Then you can drive to your area and walk, then still drive to that remote beach, remote mountain area, campsite famous place and see much more....you probably won't even pay anything for parking if in Scotland.

I would suggest if hiking is your thing get a car and look up some books on the munroes (mountains) pick which ones you want to do and make trips to them.

I did some research based on several replies indicating a rental car was the best/worst option for us (LOL, everyone has an opinion, thanks for all the PMs):

A rental car booked online here in the US and picked-up in Southampton, with full insurance and taxes, is about 160 GBP, or $ 265.00 US a week.

Comments above cause me to conclude, if I might summarize, that if you plan on visiting and traveling from rural area to rural area, a rental car is the best bet. If you are traveling from major metro area to major metro area, public transport is best.

So, if we sequence our month in UK correctly, we might rent a car for a week or two, use it to travel around the countryside, then turn it in and take public transport to visit the major metro areas, such as London and Glasgow.

Make any sense? Comments? Thanks again for all the help!

Yes it does, but could I implore nay beseech thee...ok that's going a bit far..if traveling to Scotland go up past ullerpool to the area called inverpolly, this has some of the most rugged scenery in the UK. Most people in the UK. Have never ventured up here and that's good ..we in the know have it all to ourselves ...just google inverpolly, stoer point lighthouse, stac pollard (stac Polly) this is some if not the finest walking in the country. Unfortunately you cannot get here by public transport...the area has rugged mountains, white remote unspoilt beaches, that look like white Caribbean beaches in the summer and in fact the thousands of Islands off the coast are called the summer isles ..google summer isles. The wild life here is amazing too, whales seals and dolphins off shore. In land snakes yes we have adders and slow worms, sea eagles. I spent long periods of my military life teaching mountaineering kayaking etc in this area. As you say you are military or ex military there are military connections here too with museums to the Atlantic convoys here and secluded bays so remote yet at small little information points they show pictures of US navy ships crammed in....amazing....

On another point if you message me there is a couple of veterans hotels in London only for military or veterans obviously don't want to put their details on here but they are like the officers mess very nice and in very good areas of London so you won't pay a fortune to stay there..if you want details message me.

I think the advice given on this thread is good advice.