If you like your amp, check your amp for some sort of output (many have one). Depending on what level your output is you may be able to plug directly into the tascam. If in doubt read the manual or ask a pro.
There are so many combinations of amps and pedals and processors it is hard to summarize especially without a price range. Best bet is to go in early AM (empty) to a store, with your own guitar, and see what you like.
The basic choices, outside computer stuff:
0) Guitar amp / head which just makes the sound you want; since you mostly want drive this probably exists if you can afford it.
1) Standalone pedal designed to be run into a guitar amp (i.e. an overdrive pedal like the Boss SD-1)
2) Multi effect pedal, more than one effect in a single (usually more expensive and complicated) pedal.
3) Standalone processor/effect which is capable of modeling BOTH the effect AND the amp/cabinet (essentially a higher end, more capable, "multi-effect" pedal) and which can then be run directly into a PA system, studio recorder, or your Tascam; you CAN run these into a guitar amp should you choose but you do not HAVE to do so;
4) A digital guitar combo or head, like a Line 6, which will combine #2 or #3 with an actual amp.