All new or renewed passports are now blue, and the "European Union" has been dropped. However, you don’t need to get a new passport straightaway and can continue to use your current one as normal until it's close to its expiry date. Just be aware that under new rules, when you visit most EU countries and Iceland, Liechtenstein, Norway and Switzerland, your passport will need to have at least six months left on it until expiry and be less than ten years old on the day you travel.
The rules around EHIC (European Health Insurance Cards) are changing. UK nationals living in the UK who currently have an EHIC can continue to use it in the EU until the card expires, even if that is years away. However, from January, you'll no longer be able to use your EHIC in Switzerland, Norway, Iceland or Liechtenstein, which are not part of the EU, but had previously accepted the EHIC. If you're an EU national living in the UK, your existing EHIC ended on 31st December 2020. You can, however, continue to get a new EHIC rather than a GHIC (Global Health Insurance Card), which will eventually replace the EHIC.
Taking a pet to Europe or vice versa is more complicated. Instead of pet passports, all animals now require a specific health certificate to travel, must be microchipped, vaccinated against rabies and received treatment for tapeworm one to five days before arrival in certain countries. More information can be found here.
If you want to take your own vehicle to the continent, you'll need a 'green card'. This is an international certificate of insurance issued by insurance providers in the UK, guaranteeing that the motorist has the necessary minimum level of third-party cover. To get one, contact your insurer. It will send you your green card and you need to carry the physical document when you travel. You'll also need a GB sticker (yep, like the old days). Also, if you plan to hire a car, there may be instances where you require an International Driving Permit (IDP) – like if your UK licence was issued in Guernsey, Jersey, the Isle of Man or Gibraltar.
If you were awarded funding for the Erasmus+ scheme by 31st December 2020, then you'll be able to participate fully for the duration of your exchange, even if it runs into 2021. However, beyond that the UK Government has now announced it will no longer be participating in the Erasmus+ scheme and will instead be establishing its own overseas study scheme named after the computing pioneer Alan Turing.