It's really a bad idea to be storing card details. You're opening yourself up for a world of pain in the form of PCI-DSS audits. It is not as simple as 'use encryption', you need to have processes in place to securely manage the encryption keys, schedule key rotation, securely log access and so on and on... Storing card details is absolutely something you want to avoid.
If you have to have something in place, then the best option may be for you (as a company) to take payments from the credit cards to your own merchant account, then pay the hotels separately (from your bank account/whatever). You act as a proxy for the client making the payment to the hotel.
Most payment gateways allow you to store the card details securely, and charge at a later date (using a token id returned by the gateway), which will likely be useful here. But you wont be able to retrieve the card details to pass them through to the hotel in any way, which is why you would need to take payment, then organise a separate payment to the hotel.
Its still quite an undertaking though because a lot of areas of PCI-DSS will come into play even with this simplified solution.
You asked, so here is more information:
PCI-DSS is the Payment Card Industry Data Security Standard. It's a set of guidelines which basically apply to any company that 'touches' cardholder data, in particular the card number. Touching it literally means any handling of the data, even just having it pass through your network without it ever being persisted to disk is enough to mandate that you must comply, (though it is significantly easier if you don't persist the details to disk)
You didn't yet state which part of the world you're in, or how these card details are captured (internet/telephone/in person). These details are significant to how you can achieve compliance.
Start by taking a look at the PCI-DSS SAQ (Self Assessment Questionnaires). These SAQ's are the minimum requirements for companies that do not store cardholder details to disk, and should give a good impression of the security that needs to be in place across the network and policies that should be applied across the company.
As I said, if you're thinking of storing card details then things get more complicated, because as a general rule the SAQ is no longer good enough. You need to enrol the assistance of a QSA (Qualified Security Assessor) who will visit and advise on best practice for data storage and the various other points that come into play. For this level of compliance you're looking at yearly audits (carried out by the QSA), and quarterly network scans. Take a look at the audit procedures to get a detailed look at what is involved. In particular take a look at section 3 and do not underestimate the difficulty of implementing proper key management.
In summary, full PCI compliance will be very costly. Even for a company which already has pretty strong security policies the cost of bringing in a QSA and running quarterly scans and yearly audits alone will likely cost $thousands.