Yes, definitely seal them once you've cleaned them, otherwise they literally suck dirt in from everywhere!
Re weekly cleaning, I've unfortunately found that the steam mop makes the seal wear off so I'm going back to mopping for maintenance. Have just deep cleaned and resealed and polished mine this week. It is a yearly thing. You can tell when it needs doing as it looks dirty and seems to be really dirty when you mop it, even if you mop it 3 times!
BTW, some tiles need to be sealed before laying, but not quarry tiles. We just hosed ours down, scaped any large bits of cement off, dried them indoors, got them laid and then did the deep clean to get rid of the grout as well as the original dirt.
Most cleaning products will remove any left over sealant/glaze from the tile - you then clean it as much as you can, and then reapply a new sealant. We used HG's Shine and Seal on our Terracotta tiles after thoroughly cleaning them, and they've come up quite nicely, and are much easier to keep clean now.
You need to deep clean, then seal, then polish if you like. (I have reclaimed quarry tiles in the kitchen.)
Try the HG range. Or the LTP range. Both available in good DIY or tile shops.
LTP do a product called Grimex www.ltp-online.co.uk/ltp-grimex_1.html which is good for deep cleaning. It is ammonia based and stinks when you put it on so get the windows open. I use it undiluted. You need to scrub a bit so get a deck brush to save your knees.
Does anyone have any advice about how to clean old quarry tiles? I've scrubbed one with a scourer and soap powder which has got the tile clean but seems to have left it more porous/matt. Are there any products out there that anyone has actually tried?