I need advice on my road bike tires. Background:

I never rode consistently until this past summer, when I bought a cheap, used mountain bike to commute to work. Since July, I have never once got a flat tire on the MTB. (I average about 3 days a week by bike; it's 9.5 miles each way, relatively flat).

In November, I bought a used road bike. Not fantastic, but so much lighter and faster. The problem is, I got two flats the first week of riding it.

Since then, I have mostly ridden the mountain bike, because it has been a little icy and I feel more comfortable on it. I was also annoyed at the flats.

Last week, I started riding the road bike again. Two more flats! It's a pain to change, because the tires are really tight, and I don't have much experience. Even the fourth time, it took me more than 15 minutes. And it was cold! It's now been four flats on seven round-trips. To me, that is just not worth it, even though the road bike saves time and energy vs. the MTB, and it's just more fun to ride. The current tires on the road bike look a little worn, but there's nothing obvious wrong with them. They look like most standard road bike tires. They just seem thin.

So here is my question - should I get new tires? Any recommendations for road bike tires that are less likely to get flats? Or are there tubes that are less likely to get small holes in them? Or should I just learn to live with this, and get faster at fixing/changing the tubes for flats? I am planning to go to my local bike shop tomorrow, but I also wanted to get input from people on this forum.

This is my first post, after spending a half hour looking for guidance in these threads, so apologies if this has been asked and answered many times before.

Thanks.