I am performing a science experiment where I need to generate 25 volts, 45 volts, and 65 volts. I need to attach the power source to electrodes by alligator leads into electrophoresis gel. What would be the most efficient and cost-effective way to generate these voltages.
Buy power supplies. No, really. These things are produced in large quantities and sold into a competitive market. Leave the safety design and regulatory approvals to someone else. Anyone that has to ask here won't be able to design a "efficient and cost-effective" power supply compared to commercial offerings. Even if they could, it would cost way more than just buying them unless the volumes are 100k or more.
Jameco is one place to look around since they have a very wide selection of power supplies, and they target hobbyists and other non-professionals.
I believe your current requirements will be modest for electrophoresis.
Consider a string of 9V batteries, which will be inexpensive ($7 at the dollar store for Panasonic batteries) and relatively safe. If you can add a 100K resistor to the taps they will be even safer, but avoid contact with the batteries.
It's possible to regulate the output of the batteries and do other fancy stuff, but if you just want roughly those voltages you can tap them all off of one string of 7 batteries (eg. 27V/45V/63V).
There is also an instructable specifically to make an electrophoresis supply that appears to be above par for the genre, and the PCB is open source so you could easily duplicate it on a similar PCB or on perf board. The voltage is regulated and is adjusted with a pot. Assembling a dozen or so parts is not exactly a Herculean task if you have wielded a soldering iron before. It's based on the MAX1771 chip (available in easy-to-use 8-pin DIP form) in a standard 'boost' configuration. You can power it with a 12V/1A wall wart adapter.
I would feel more comfortable if you are using a power supply (such as the above suggestions) with little output current capability and with limited output voltage. I have a 400VDC 2A programmable power supply kicking around here, but interns/juniors don't use it without a lot of safety instruction and preferably not working alone.