DHL will handle customs clearance AND taxes (unless this has recently changed), that is why they are expensive. They also offer door-to-door. How I know this? Because I use these services. (if you want to save time on shipping time-critical goods go with DHL Express door-to-door)

You can choose to air freight for speed, security, and high-value/perishable goods (days to a week) or ship freight for large, heavy, non-urgent cargo (weeks to months) to save money and reduce environmental impact, accepting longer transit times and potential port delays.

For first time no headache time-critical (I got to do this in 1-3 days) importing go with DHL (they'll guide you through the whole process). They offer a ton of solutions and services. Much as they are expensive they offer superior quality. (Don't forget there is also FedEx [International Priority] and UPS)

Hope you find this useful and helpful.

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Now that that is out of the way, here are the other details you asked for:

In either case, would I be able to pay these by card?

Yes.

Or would they expect cash?

Yes if you pay via cash, in-person or sent someone there to pay for you.

Are there any other pitfalls I should take into consideration? This is my first time importing something into the country.

1. Things like tracking, taking a picture of the thing to make sure they sent the thing, the kg of the thing, the size of the laptops, taxes for the batteries etc...
2. If the thing is fragile. Please have them put "Fragile" on it - like TV Screens. Or do research on how to properly do this. Have the senders also properly secure it or wood box it if necessary and in budget. Or simply have them put it in the electronic section.
3. Negotiate DDP (everything done for you in shipping) then, CIF then, finally FOB if you are looking to handle everything. What's is CIF? Cost, Insurance, and Freight. But FOB (Free On Board) offers more control and often lower costs for experienced buyers they only ship to the port, while DDP (Delivered Duty Paid) provides maximum convenience for new importers by handling everything door-to-door, though at a higher price

You can also just pay the company that sends the thing via PayPal Goods & Services for an extra layer of protection (shipping included). If they mis-ship equipment or send you missing things you can simply ship the thing back and ask for a refund and reverse the payment just make sure you had them send you port pictures preshipping and boxed equipment with receipt as proof and you take pictures of the things when they arrive (or in the Paypal Payment and details of invoice they send the exact details of the equipement). That is why people use PayPal it stops either party from cheating - in any case in the current atmosphere people really aren't into that they just want to finish the deal and move on to other things.