How to do 'modular-interconnecting' board sections?

If you are in the early stages make the pieces loose you can always do more with them latter. I suggest getting a rubber mat as those do a really go job of holding pieces in place.

I think Jigsaw will work the best, however it require precise cutting. If the cutting is less than perfect the entire system does breaks down and starts to fray and damage the pieces get damaged. so I recommend Jigsaw if you have the machinery to do it right.

I do not recommend plastic clips. The are annoying to put together and generally damage the pieces. I also do not feel they work that well.

Actually, I've played enough board games where there are tiles of thick card simply placed next to one another without any connection, with game pieces plonked down or slid over them, that I don't think you need to worry too much about this unless you're really moving huge numbers of pieces back and forth.
But here are a few unorthodox ways in which you could make a connected modular board anyway:

As a child, I had toys where you created a scene by pressing thin vinyl cut-outs (eg. with human and animal figures printed on them) onto a smooth plastic backing with some printed artwork on it (eg a zoo or farm). The vinyl stuck pretty well, but could be peeled off and re-used pretty much indefinitely because there was no glue involved. I haven't seen these toys around recently, and I can imagine that printing quality on glossy vinyl might be slightly lower than printing on card (texture would also suffer a bit I think).
Felt-backed tiles on a felt board will also stick relatively well I think.

Alternatively, if you're willing to go to the extra expense in your components, you could use flexible magnetic tiles almost like some fridge magnets (then if you distribute your game in a metal tin, it can stick directly to the base and/or the lid - my son has a toy almost exactly like this at the moment, so they are being manufactured). As a bonus, this would be great for a travel-friendly game, but rather limiting in terms of play-area size. You could also try something exotic like velcro-backed pieces and base, but I'd hate to know what the cost of production will be.

Absolutely go strictly with loose tiles laid out on the tabletop.

There are MANY games out there that use this method and it works without the fiddly bits needed to clip together or the damage caused by the jigsaw puzzle method .. both of which add more work to the gameplay.

Add on to that the cost savings in production and it's the only choice really.