There is no upgrade path from 32-bit Windows to 64-bit Windows. You have to wipe the drive and reinstall from scratch.
The main benefit of 64-bit Windows is the ability to use more than 4GB of RAM. A smaller secondary advantage is that 64-bit calculations (very rare outside scientific circles) can be done more quickly. If you have 4GB or less RAM, there's practically no advantages and some disadvantages (have to wipe and reinstall, 64-bit programs use more RAM). So you should just stick with 32-bit Windows.
Long-term support for Vista ended earlier this month. That means Microsoft is no longer patching security flaws found in Vista. It's gradually going to turn into a malware cesspool, so I would highly recommend upgrading the laptop to a newer version of Windows, or Linux if you can't afford it. A retail Windows license works for both 32-bit and 64-bit versions, so if you get a new copy of 32-bit Windows for your laptop, in a few years when you upgrade to a new laptop you can re-use the license on a copy of 64-bit Windows for your desktop. Just make sure it's a retail copy, not the cheaper OEM copy. You are only allowed to install the OEM copy onto one system - you cannot transfer it to a different system in the future.