Any new paint you apply will not adhere to the varnish - it's too smooth, so you will definitely need to remove it by either chemical or physical means. As you need to introduce a *rough* surface, you could sand the old varnish down (thus providing what is called a key). Some people use chemicals to remove the old varnish - it's up to you which method you choose (it's quite hard to get into all the nooks and crannies with sandpaper - you may need to use a special little scraper). Both will need good ventilation (sanding down produces a lot of dust and the chemicals produce some smelly fumes) so keep kids and pets away whilst you work - you may wish to wear a breathing mask. Once you have either sanded down or chemically stripped the old varnish off, wash the railings down with some water and allow to dry thoroughly before undercoating.
It's a good idea to use a good quality undercoat, then a top finish coat or two, all applied in very thin layers, so you get a really tough hard-wearing finished surface. Use gloss paint if you want a shiny finish, or eggshell if you prefer a slightly matt effect.
PS This is a lot of work but the secret is in the preparation - the final painting will only take you an hour or so, whilst the sanding down/removing of the varnish may well take all day. The more time you can spend on the prep work, the better the finished result will be.