If your whole reason for going to Hong Kong is to see DL, then stay on-property, which I think you've already booked in anyway. I highly recommend you take a taxi from the airport to HKDL. It will cost about HK $130, which is about $18 or $19 in US money. When you share it 4 ways, it is excellent value and you won't need to worry about bags on buses or trains. Take the Green Taxi.
It will take roughly 30 minutes by MTR to get from Disneyland into Hong Kong city. The MTR service in HK is excellent. It will be cheaper than taking a taxi from HKDL to the city. As a guide, it might cost about HK$250 per taxi trip, roughly $35. The MTR will cost about HK$10 - $15, roughly $2.
There are lots of things to do in Hong Kong. It just depends on how much time you have.
Someone has mentioned the Big Buddha. It is on the same island that HKDL is on. To get there from HKDL, you can catch the MTR to Sunny Bay and tranfer over to the Tung Chung line. Tung Chung is the last station on that line. It is less than 100 m to walk from the Tung Chung station to catch the cable car to the Big Buddha. I highly recommend the crystal cable car cabin.
Your alternative is to take a taxi. I'm not sure how much it will cost.
The various markets in Hong Kong are just wonderful. Stanley Market, and Stanley is fantastic. I caught local transport (bus) out there...either the 973 from Canton Road in Tsim Sha Tsui or the 6/6X bus from Central station will get you there. But I will say that the price of goods at Stanley Market was more expensive than at the Temple Street Market or at the Ladies Market. And that was without haggling! I did buy some jade trinkets at Stanley Market and it was a couple of Hong Kong dollars cheaper at the Jade Market. But for the experience, I was glad to pay the difference.
I'm usually very cautious about where I use debit and credit cards.
We predominantly used cash when we were in Hong Kong. I did bring out my credit card twice; once in Sogo and once in Zara. I was worried about the CC in Sogo because the server took it away and out of sight to process it behind closed doors. If that worries you, then just use cash. You'll need cash if you're shopping at the markets anyway. I would not want to use any sort of card; even if they did accept them.
We have a debit travel card that we bought HK $ when the exchange rate was good. I'm not sure if the banks in the US offer this service; but I think it should. We took out money at the HK bank ATMs without any issue with the debit travel card. We also took money at the HK bank ATMs without any issue using the debit card linked to our Aussie savings account.
I didn't travel with traveller's cheque to HK. I'm not sure how widely accepted it is there and you might find that you'd need to find a place that would cash it for. I expect that the hotel concierge would do that.
If you let us know how long you're in HK for, we might be able to provide other attractions to go see.
(I do have a TR going on my time in HK and will probably be posting pictures and my views on most things. I should have it done before June...maybe. I might even have pictures of the Hollywood Hotel to share at some stage. 
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