Do you have any suggestions on reliable companies that will transfer myself and husband to our hotel in Seoul?

There is an easy bus from the airport into Seoul; whether it would be convenient for your hotel depends on where you are staying. If that doesn't suit, I would think your hotel should be able to offer an option.

I was thinking of taking a taxi service. Do you have any suggestions?

https://www.tripadvisor.com/Attracti...562-Seoul.html

I looked it up. As you can see a shared ride in a van is pretty reasonably priced. Private vehicles are also reasonably priced. It’s certainly not like Japan.

I also saw that Uber is available too.

Ascmentioned above, there are lots of reasonably priced options at the airport. No need to book. Never a fan of Uber from airports as it always seems the airports try to make it as difficult as possible by putting the pick up point in hard to find locations

Last time there I was assured that public transport was an easy and cheap options we got the airport bus. Getting off at the rights stop was the first challenge. Second was finding the right subway line. Third was finding the right exit out of the subway station! Taxi is a good decision ! Wonderful city BTW

First, check if your hotel has shuttle service from and to the airport. If they don't, consider these aspects: budget, number of people traveling, kids / elders, distance from your accommodation / hotel to bus stop / metro station, time of your arrival, and so forth. If you're traveling as a family and have flexible budget, you can choose jumbo taxi, ride hailing service, or private transfer. Otherwise, go by bus or AREX train.

We used Transfeero from ICN to our hotel. Was about $60-ish, everything went smoothly.