My family and I visited Dublin, Ireland & Edinburgh, Scotland in November of 2013, and the weather was about the same as what you will experience in March: Cool to cold and cloudy with misting rain a lot of the time.
It was my fifth trip to Ireland and fourth trip to the UK. My wife had previously visited Ireland & the UK and our son had visited the UK & continental Europe multiple times with us but had not been to Ireland. My parents (who went with us) had been to the UK and continental Europe many times before but not to Ireland..
We flew from the USA to Dublin on Delta. We visited Dublin for a week and flew to Edinburgh using Aer Lingus. When our week in Edinburgh was over we flew back to Dublin for an overnight, and flew back to the USA the next morning.
In Edinburgh, we stayed at a hotel near the city center:
We did normal tourist things, and another answer is correct you should look through a guide book.
You can travel to London by train and then go to continental Europe via the Chunnel train. It is not cheap or quick.
We had no problems with the food anywhere. We avoided tourist trap restaurants so paid normal prices. Most restaurants have a menu posted outside the entrance so you can see what they have and the cost before entering.
The best ways to minimize costs are to book in advance (now is about right) and not change locations any more than necessary. Changing from one city to another always burns up a whole day of your trip, and there are costs such as transportation between the airport and the hotel every time you change. Also choose a hotel that lets you use public transportation or walking to get around. Staying way out in the suburbs to get a lower hotel rate is not a bargain if you have to spend money on taxis for a long ride to the city center every day.
Some general hints:
You need a passport and you should apply not later than next week. You apply in person at the post office:
Get a passport card in addition to the regular passport. Don't keep the passport card with your passport (so they can't both be lost/stolen at the same time):
The cards are only valid for crossing land/sea borders within North America, but they make great daily ID when traveling abroad. That lets you keep your regular passport (and US driving license) safely tucked away.
The cards also don’t show your home address (like a driving license or state ID card does), making it much less likely someone could steal your identity while you are traveling.
Check with your health insurance to find out if it covers you outside the USA and pays for medical evacuation to home (should that be necessary). If yes then take proof of coverage with you. If no or not sure then get trip medical insurance, which is cheap and sold by airlines & travel agents. PLEASE do not skimp on this as an otherwise-silly accident could turn into a crisis if you don't have medical insurance.
US citizens do not need a visa to visit the UK as a tourist and you should NOT apply for a visa.
You need to start looking at airline tickets NOW.
Once you decide on which airliner you want to use then create frequent flier accounts for both of you. Log on to one of the accounts to buy the tickets.
You also need to start looking at hotels now. Same thing with frequent guest accounts.
PRINT copies of your airline itineraries and hotel reservations for both of you. The copies should go in your carry-on bags.
Scan the photo page of your passport and the front of any other IDs you will take with you and send the images to yourself via e-mail. Scan the fronts & backs of any credit/debit cards and send the images to yourself. That way the information is available via any computer connected to the Internet.
You should get about 100 Pounds in British money before leaving home. Get it in 10 Pound notes. Leave the rest of your money in your bank account and use your debit card to get cash from ATMs as needed. Call your bank and let them know you will be using the card in Europe. Ask if they have a partner bank in the UK so you can use their ATMs for a reduced fee. In any event, only use ATMs which belong to a real bank.
Don't take anything that requires a voltage converter. Most electronics made any time recently work on any voltage and all you need is a UK plug adapter. Look on the power supply or device: If it says something like “Input 100v-240v” it works on any voltage. Adapters are sold at electronics stores and luggage shops (or the luggage section at a large department store). Electrical devices like hair dryers and curling irons tend to not work properly, even with a converter. Borrow or buy electrical appliances after arrival, and they are sold everywhere.
Call your USA cellular provider and set up your phone so it works in the UK. Using it there may or may not be expensive but having the option can be important. Know what to dial to use the phone in Europe.
Use Magic Jack or Skype to call the USA. They both require a decent Internet connection but calling the USA is either free or very cheap.
Take one carry-on bag each. Your gf can have a purse in addition to the carry on bag. Pack basic toiletries and one change of clothing in the carry on bag. You are allowed to pack "travel size" liquids & gels in your carry on bag, but the items have to go in their own clear plastic bag and scanned separately when you get to security.The rules are on: www.tsa.gov
Take aboard the flight a hooded jacket that is water resistant, a sweater, a knit hat, gloves, and perhaps a folding umbrella.
Use one normal size rolling suitcase which weighs a maximum of 50 pounds when packed. Clearly tag ALL bags >>> inside and outside <<< with your name and a good phone number + e-mail address (but not your home address). The free tags from the check-in counter are OK, but don't forget to do it..
I hope you have a good trip!