Is there a way to download mods without WinRAR? It is quite complicated for me to download even one mod with WinRAR, such as Rei's Minimap Mod and TooManyItems Mod.
Yes, it is.
You need to download directly from mod author's websites, those nearly everytime come as JAR files, so all you need is to put them into MODS directory of your installation. You don't need any WINRAR work with them. They are not meant to be extracted by any means.
Try to avoid those unoficial websites with "all mods here". Use Curse, search for official Minecraft Forum thread of that mod or official author's page.
If you had in mind that you need to open minecraft.jar with WinRAR and drop mod files into it, you should know this is not needed anymore and it is strongly discouraged. By any means, you should keep minecraft.jar untouched. It was a thing years before, but for new versions you just need to use Forge or similar (LiteLoader), then just use MODS folder for the mods. Also mods themself do not support any other behaviour anymore.
That is why you don't need to use WinRAR anymore.