Yes. It used to be exceedingly expensive to use mobile broadband abroad, but these days the costs have come down. Check with your network how much you will pay. Some require you to pay a fixed fee per day and others have a usage charge based on what you use. In Europe the costs can be a lot cheaper than in other countries due to European regulation. In 2014, European costs should drop to be the same as when using your device in the UK.
Make sure you check with your provider on the charges for the country and network you're going to be roaming onto. Some require you to active this before you go away.
Remember your inclusive data allowance does generally not apply abroad.
You can use your UK broadband modem abroad, but most of these will be 'network locked' so you can only use them with your UK SIM card (and therefore have to pay the UK charges). This means you probably can't use them with a local pre-pay SIM card. Try to make sure your next laptop is one with a SIM card slot and mobile broadband built in as these tend not to be network locked. Also, be aware mobile broadband can use different frequencies in some countries so check with your provider if you are unsure
Yes, if you are using 3 Mobile Broadband, you can use a PAYG modem abroad provided you activate your account for international roaming. For others, you need a contract based broadband package to use it abroad. You may find that some month-to-month contracts allow this, but it's very high risk and you could end up with a large bill at the end of the month.
Prices and deals are often changing. The best value option will often be where you can use your existing UK package for a set daily fee. Check with your network operator before going abroad to find out how much it is likely to cost. Most now have warnings if you are approaching a set limit and may also have usage caps.
Make sure you don't download any operating system or application updates when roaming as these can consume a lot of bandwidth. Also, avoid YouTube, iPlayer and any other video content. It is not unheard of for mobile operators to send bills running into thousands of pounds for some use so don't take any risks.
Check out local pre-pay mobile broadband SIM cards. Some plans for phones may not work as fast, but at least with PAYG you are protected against unexpected charges.
Consider using wireless hotspots (Wi-Fi); most laptops support it and many hotels and cafes have free access points. Do be careful not to pay over the odds--Some hotels charge over the odds for broadband access.
Internet cafes are great if you don't want to bring your laptop, but make sure you are careful where you surf. We would advise against carrying out online banking or shopping transactions from Internet cafes, and would recommend at the very least you change your e-mail/Facebook passwords when you get back if you have used these from an Internet cafe.
If you're on holiday, consider taking a break from the Internet :)