It's as well to use sanding sealer which will fill the grain, prior to using varnishing. Sanding sealer is basically lacquer with zinc stearate added, which is a 'soapy' substance which fills the grain. The zinc stearate settles to the bottom of the container, so needs a good shake before application. You can apply it with a brush or cloth.
It comes in both cellulose and shellac - the cellulose type dries quicker, but the shellac dries in a matter of minutes anyway, then you can lightly sand it before applying whatever varnish you wish to use. As you say, modern veneer is very thin - usually just 0.6mm so you need to be cautious about sanding. It's become rather expensive, especially in small quantities such as 250 mL and if ordered by post.
Chestnut and Liberon brands are well regarded. EG:
https://www.ebay.co.uk/itm/Chestnut-...Sn60fJ47ATESYA
Hope that helps.