From Warsaw to Wroclaw with kids?

Sorry, my english isn't very good...

We are going to visit Wroclaw next week. Our flight will arrive Warsaw airport at 8.30 on Sunday morning. What is the easiest way to get to Wroclaw? I think the cheapest is by train? We should take taxi or bus from airport to railwaystation? The biggest "problem" is that we have two little kids (ages 2 and 3) with us. It won't be very easy to carry kids and luggage to the bus and to the train... Can little kids travel free by train? LOT flight from Warsaw to Wroclaw isn't very cheap. Is there any cheaper flights? How long it takes to travel by train from Warsaw to Wroclaw? It would be nice if someone could give me some tips travelling with kids in Poland.

Bus 175 is the cheapest way from the airport to the railway station.
Taxis are tricky: there are two kinds of them, the cheaper usually are from Taxi Corporations (you can see they have short telephone numbers like 9222, but someone who lives in Warsaw could tell you more, because I visit that city rarely and in my own car, so I don't use taxis), there are also more expensive "freelance" cabs, their fares can be much higher, better to avoid.

Polish railways in English:
rozklad-pkp.pl/bin/query.exe/en
(also German and French language exist)

on the image attached there's a frame (grey background, bottom of the page) it indicates the distance (471 km), and prices (129 Polish złotych for 2nd class, 180 złotych for 1st class) for this specific connection (if you choose other trains the price will be different), unfortunatelly, I don't know how old a child can be to not have to pay for the ticket.

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I certainly wouldn't take two small children and luggage on the 175 bus. Because it's the airport bus, it attracts a lot of theives and pickpockets. On three trips I've seen people being robbed on that bus. An authorised taxi will cost you about 35zl on a sunday.

As far as I know, no airlines other than LOT fly from Warsaw to Wroclaw. The reason LOT put the price of Warsaw to Wroclaw flights so high is that the train connections from Warsaw to Wroclaw are so bad.