Well, I would like to present some different, than Mark's & Piotr's, point of view. You have plenty of time, so you can get from Auschwitz to Wieliczka by bicycle and visit both KZ and salt mine in a single day :)

tl;dr

Yes, it is possible to do the single trip with Auschwitz and Wieliczka in one day!

Hiring a car, taking an organized trip, or just get Uber/Bolt/Freenow/cheap taxi (see below sections) would be my recommendation. Public transportation would be cheaper but yet more challenging option. It requires more free time and take-ins, but it is not impossible.

You may also get a bicycle trip. Having 5.5 - 8 hrs. spare time (depending on the month) and 80 km distance, you can get to Wieliczka by bicycle.

NOTE that Auschwitz in Polish language is O?wi?cim. In Poland, Auschwitz name is considered more as the concentration camp and O?wi?cim as both camp and the city.

Auschwitz-Birkenau concentration camp

The Auschwitz concentration camp opening hours varies month by month and are:

• 8:00 AM - 2:00 PM December
• 8:00 AM - 3:00 PM January, November
• 8:00 AM - 4:00 PM February
• 8:00 AM - 5:00 PM March, October
• 8:00 AM - 6:00 PM April, May, September
• 8:00 AM - 7:00 PM June, July, August

Also note:

These are the hours of entrance to the Museum. A visitor may stay on the site of the Museum 90 minutes after the last entrance hour (i.e. 5.30 in February or 8.30 in July)

Time needed for Auschwitz-Birkenau concentration camp is 3.5 hrs:

As a minimum, however, at least three-and-a-half hours should be reserved.

Wieliczka Salt Mine

Please note that Wieliczka has a single salt mine. Another salt mine open to sightseers in the region is Bochnia Salt Mine which is located slightly more than 40 km (~25 mi) from Kraków.

Wieliczka Salt Mine opening hours for the "Tourist Route" are:

The route is open from 09:00 am to 05:00 pm, and the tour in English-speaking takes place at the following times: 10:00 am, 11:00 am, 12:00 pm, 01:00 pm, 02:00 pm, 03:00 pm, 04:00 pm, 05:00 pm

The closing hour means "hour of closing ticket desks and last group of tourists entering the mine."

Please note that alternative routes (Miners' Route, Pilgrims' Route, Group Adventure, The Mysteries of the Wieliczka Mine) have different schedules.

Planning

I'd suggest to start with Auschwitz-Birkenau concentration camp.

Starting from 8:00 AM + 3.5 hrs = 11:30 AM.

Assuming the last entry to Wieliczka Salt Mine is either 7:30 PM or 5:00 PM, that gives you consecutively 8 or 5.5 hours of time (depending on the month) for getting from Auschwitz to Wieliczka.

The distance from Auschwitz to Wieliczka is approx. 80 km (~50 mi). The route can be taken via "A4" highway and it takes, in the optimistic scenario, 1 hour. You may use Google Maps which calculates pretty accurate traffic in Poland. I wouldn't expect much traffic in the noon/afternoon time.

Get In

? Car

Hire one and you will have a plenty of free time. Remember to use Google Maps to avoid tolls and traffic.

?/? Bus, Minibus & Train

Whether you go by bus, minibus or train, if you want to check timetable online, you should probably look for:

• From: Kraków \ Kraków G?ówny \ Kraków Dworzec G?ówny
• To: O?wi?cim

That may be obvious, but some may want to look for Auschwitz rather than O?wi?cim.

Direct and indirect connections

Go to the e-podroznik (English language available) and check the route for your destination. There are many options available and the website lists the travel time, walking time, transport type, prize and option to buy a ticket online (for some routes).

From-To Auschwitz

Schedules:

• Train / bus / minibus: e-podroznik (recommended one)
• Train / bus / minibus: http://www.rozklady.com.pl/
• Train: http://rozklad-pkp.pl/en/

As per Auschwitz Memorial & Museum official website:

The Museum is located on the outskirts of the city of O?wi?cim on national road 933 (note: entry to the parking lot is at Stanis?awy Leszczy?skiej Str. no. 16). The Museum is about 2 km. from the train station and can be reached from there by local buses.

Transportation between camps is as following:

Communication between former Auschwitz I and Birkenau concentration camps

Auschwitz I and Auschwitz II-Birkenau are situated 3,5 km from each other.

Car parks

There are paid car parks available near both former concentration camps.

Museum Bus

Every hour the museum bus shuttles from one concentration camp to another. Departures from the former Auschwitz concentration camp: between April - October every 15 minutes, between November - March every 30 minutes. No bus fare is required.

TAXI

Taxi services are provided by HALO TAXI company. You can find a taxi at the taxi rank in front of the Museum entrance. You can also call a taxi using the following phone number: +48 32 19 194.

By foot

The three kilometre distance between the former concentration camps can be covered by foot (through streets: Leszczy?ska and Wyzwolenia), crossing the by-camp premises, where during the occupation were German industrial plants and workshops, warehouses, offices and technical backrooms, in which the prisoners would work and die. There are preserved remains of a few railway sidings and ramps [link to Judenrampe], where the trains with deportees would stop and the members of the SS would make the selections.

From-To Wieliczka

Schedules:

• Train / bus / minibus: e-podroznik (recommended one)
• Train: http://malopolskiekoleje.pl/
• Bus ? 304: http://www.mpk.krakow.pl/en/page-70cdbe44/
• Other buses: http://krakow.jakdojade.pl/?locale=en

As per official website:

Almost all underground routes begin at the Danilowicz Shaft located in the area of the Mine. Only The Miners’ Route and the Mysteries of Wieliczka Mine start from the Regis Shaft located in the centre of the town.

? Organized Tour

There are many organized tours in Kraków, so you may easily reach one and ask for a single day tour.

Example: Cracow City Tours organizes a full day trip to Auschwitz and Wieliczka.

? Taxi, Uber, etc.

You may also try Uber, Bolt or FreeNow.

I've just checked Uber and Bolt options -- it costs around 275 PLN to get from O?wi?cim to Wieliczka.

If not, you may try to call either iCar or Mega Taxi, which are the cheapest and the most reliable taxi options. Both taxis have mobile apps, so you may use these as well.

???? Bicycle

It's around 80 km to get from Auschwitz to Wieliczka, so... why not biking. Having 5.5 - 8 hrs. spare time (depending on the month) and 80 km distance, you can get to Wieliczka by bicycle.

You may use Google Maps for the bicycle trip, but Google Maps may do not find suitable path for your bicycle type. I'd recommend to use either Mapy.cz app (free option and you can choose between Mountain Bike and Road Bike) or Naviki (free with minimum options, but you can choose between plethora of bicycles: Everyday/Leisure/Mountain/Racing/S-Pedelec).

With avg. of 25 km/h, you will get to Wieliczka in less than 3.5 hours.

? Costs

Also, I'm wondering if it will be cheaper and more cost-effective to join to an organized group or do it on my own?

Organized trip would cost you about twice as non-organised trip with public transportation. The cost of a single-day organized trip with tickets included is 260 PLN (~60 EUR). An entrance to Auschwitz-Birkenau concentration camp is free. An individual ticket for Wieliczka Salt Mine costs from 80 PLN (~20 EUR). Public transportation requires more take-ins and may be much more difficult, but still – it is doable.

It's not easy and not recommended to do both in one day!

The best is either to have the private car or to choose the rush tour starting very early morning:

The tour of Wieliczka Salt Mine takes at least 3 hours. The location of the mine is half an hour drive by car from the center of Krakow. The town of Oswiecim (in German: Auschwitz) is about at least 1 hour and 15 minutes drive - better to say 1,5 h drive by car from Wieliczka (the highway is distant). And you need there at least 2hours to see the most important parts of Auschwitz KZ.

You should add an extra 1h for traffic jams (very common in Krakow) and it is: 3,5-4,5 hours of drive + circa about 5-6 hours for both tours (it's not easy to shorten them, especially in Wieliczka Salt Mine where is almost obligatory to have the reservation for the entrance). So it can be a tiring 10 hours without lunch and stops. Let's say 11-12 hours.

As users before suggested - it's quite difficult to make it in one day, but it is possible.

From Krakow to Wieliczka it's not very far, but there can be traffic. Traffic between Wieliczka and Oswiecim is also quite "difficult", and it would be more then hour one way. For Wieliczka - You need about 3-3,5h with guide only, for Auschwitz more or less the same time, and it's also suggested to have a group, toured trip. Summing the fact, that You need to move around quite a lot, and have 2 guided tours, did you consider ordering a day service? It may be the better way in your situation (low on time, lot's to see :) ).

Good luck!