MacBook Pro (2008?) running Lion 10.7.5 - how its 'obsolete' status will affect my photos etc
Here's more information on vintage and obsolete products that may be of interest to you.

Your questions actually covers two main concerns. I'll try and address these below:


Web browsing: Many users of older Mac hardware and/or versions of the OS do tend to install additional browsers in order to be able browse more modern websites. Firefox in particular is useful for this, but others use Google Chrome. Either way, this addresses the websites acting funny issue in most cases.
Photos: In terms of iPhoto, this was superseded by the Photos app which was released with the OS X Yosemite 10.10.3 update in April 2015. In most cases users have just converted their iPhoto Library to become the Photos Library. In some cases, users still maintain both (although there is no need to do this - not only is it doubling up on valuable storage space, it's also a pain to do). Regardless, my point is that your iPhoto Library will not be lost because (1) it can be converted to the Photos Library in future and (2) you have Time Machine backups of your data. So, unless you have manually set your Time Machine backups to not backup your iPhoto Library, there's really no need to manually export all your photos from iPhoto.


Your MacBook Pro

As an aside, I wanted to add that all 2007-2008 era MacBook Pros are capable of running Mac OS X versions up to El Capitan 10.11.6. If your MacBook Pro is indeed one of these you may want to consider whether you'd like to upgrade your OS so you can have the benefit of a more modern browser and software such as Photos. In this way, when you do eventually upgrade your hardware, the risk of any software incompatibilities will be reduced.

Of course, before upgrading your OS you would need to confirm your model of MacBook Pro and its hardware configuration. You can do this by:


going to Apple > About This Mac to view your processor and memory details, and
to get the exact model, click on the  More Info...  button and look for the Model Identifier (it'll be something like MacBookPro5,1).