Everything else equal, using a laptop vs a desktop should be very similar. The biggest difference between a very obsolete desktop and a new laptop will be the operating system, which can be quite different, depending on which you were using. You can get a mouse to use with the laptop, if the track pad is awkward to use.
Email security practices will always be the same, no matter what device or operating system is in use. If an email looks suspicious, don't open it. Delete it, get rid of it so it is gone.
If you are still within the return window for the laptop, some advice on buying one. Assuming that you want it to last a while and the primary function will be web browsing, buy the best you can afford but minimum specs should be an Intel Core I3 or AMD Ryzen 3 for the processor, and M.2 SSD and 16GB of ram (or 8gb but ONLY if the ram can be upgraded in the future. This kind of upgrade is not possible on all laptops).
For only web browsing, try to avoid laptops with GPUs (Graphics Processing Units). In that case, a GPU is only another component to fail and need maintenance. The only place they are really useful is for gaming or productivity like video editing, CAD, graphics design, etc. Otherwise, it is a largely unnecessary expense.
With lesser specs, you will probably need to replace the laptop in only a couple years, as software continues to become more sophisticated, resulting in the computer becoming more sluggish. This can make a cheaper computer more expensive over a longer time-span. If you spend a little more now, you can get a laptop that might last 5-10 years if you use good antivirus software along with common sense with e-mail and web browsing, to avoid having malicious software installed that can dramatically reduce the usefulness and lifespan of a computer.
With most laptops, despite their nomenclature, your lap or any similarly uneven surface is not the best place for it. Try to always place it on a hard flat surface so that it can cool well. If you consistently place it on your lap or carpet or a couch, etc, you will significantly decrease the life of the computer because it will run hot and cause failures to occur much sooner.
Being able to upgrade the RAM is probably the most important. With that feature, you can keep a computer functional for web browsing for a pretty long time. In fact, your obsolete desktop may still have some life in it if it does not already have the maximum amount of ram installed. An SSD would also help but that will require re-installation of an operating system, so you would need to find someone locally to help with that. Of course, if it is old enough, upgrades will largely be a waste of time.
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