That bike looks fine. As someone above has said, it is not unlike the early, rigid MTBs.
I took my wife's (now my daughter's) Claud Butler Cotswold shopper off-road many years ago, It had road tyres, but my only issue was with the mudguards, which your bike lacks.
For most off-roading in all but the soupiest conditions, you really don't need a lot of tread. Your tyres look fine for most stuff if taken gently. If you do want to go for knobblies, pay attention to the width of the tyre. There needs to be room for the wheel to turn and (when off roading) a little extra room for mud, leaves and gubbins.
And.. I wouldn't get sucked into buying some SUPER KNOBBLIE tyres that look like they fell of a moto-crosser. Just a little bit of tread that will hold where they need to. There are some which have no 'knobbles' (?) on the centre line. they may be a good compromise.
A caveat: Do not be surprised if things like the head bearings and wheel bearings take a bit of a hammering off-road. It won't be fatal, but it will take more out of them than tarmac does. Also, your chain, cogs and brakes will wear faster because of of the shoot and derision that flies around on trails.
That bike looks well up to the job though... Enjoy yourself!