I went to the Egged Bus site to see how long it would take for you to do this. (
It revealed there are no direct routes to Ashdod. You have to change buses. The trip can range anywhere from slightly over 1 hour to 1.5 hours. It cost--again depending upon route approx. 45 shekels. ( 3.75 shekel to the $$ yesterday)
Ashkelon can be reached by one bus. It takes 1.5 hours and cost 53.5 shekels. Whether you commute back and forth--or spend the whole week at each location is up to you.
I did look into going between Ashkelon and Ashdod--and that is THREE buses. Personally I'd spend a couple of days in Ashdod, a couple of days in Ashkelon ( leave the bulk of your luggage at your son's) and then go to Tel Aviv.
Egged runs the #300 bus to Ashkelon.
There are also trains to Ashdod and Askelon. These are also listed on "otobusim" or you can look at the railway web site:
Correction to rdglady post. There is regular bus service (every 10-20 minutes) to Ashdod from Tel Aviv, it takes about 50 minutes, depending on traffic. There is are also regular buses between Ashdod and Ashkelon. The train is convenient from Tel Aviv to Ashdod and Ashkelon but a bit more expensive. I would stay in Ashdod/Ashkelon to safe time and money and get to explore both towns which have their own interesting spots that many don't visit.
For greater convenience you may want to stay in Ashdod or Ashkelon. If you want to enjoy Tel Aviv in the evenings, etc. then definitely, stay there. Transportation from one to the other of these three cities is convenient, easy, and frequent. If you stay in Tel Aviv then I would suggest you base your decision on bus or train based on where your relatives live in Ashdod or Ashkelon. The train stations are on the outer edges of either town, while the bus stations are in the center of things.
If you choose to stay in Ashkelon or Ashdod and want to take a day trip to Jerusalem that is easy too - Egged has direct buses from either city to Jerusalem. From Ashkelon its the 437 (ride is about an hour and a half), from Ashdod there is an express 448 and also a 438.
Mbgg--thanks for the addition. Even though Egged showed a way to get to Ashdod--I should have checked with otobusim--since there are so many other companies nowadays. Glad you were able to make what looked like a horrendous journey better!
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