2002 trailblazer won't turn over or start

Check the cable connections at the battery and the starter for corrosion.
You may want to use a volt meter ( DC Volts ) for a more accurate reading.
Put the black lead on the battery NEG.,next ( with the meter on ) touch the POS. post/bolt with the red lead and see if you have atleast 11.0 volts, if so, try to bare a bit of the rubber insulation at the battery end and recheck with the red lead. Sometimes the solder weakens which causes a loss of voltage/current.
If reading is still close to or above 11.0 volts, next touch the red lead to the large post on the starter and see what the reading is,if this is still above the 11.0 volts,,,, check the fusable links located near the starter, you must have the same reading on both sides of the fuse ( round black plastic ).

If the battery is full as you say, your dash lights come on intermittently, and your connections are clean then you might have a bad starter switch in the column.

But try this first. This is free.

Possibly something is causing a short in the system when you try to crank it. The most likely would be a sticking solenoid or stater. Try tapping on the solenoind and starter with a small hammer while trying to crank it. No go? With the car in park/neutral, emergency break on and the key off, jump the starter from the solenoind. No go? Directly power the starter with jumper cables to see if she will even turn over. If it does not, remove the starter and have it tested and replace it if bad. If not ba, turn the car over by hand from the idler pulley or crankshaft to make sure the engine has not locked up. Check all your connections including the line from the battery to the solenoid, all grounds (engine block) and any solderless connectors like from the solenoid to the battery.

Check for corrosion between cables and battery --if it is a side post you will have to remove the cables and clean the connection with wire brush--pooring soda over the connection will also help wash away some on the corrosion--with a volt meter check voltage at battery at terminal and at cable should be very close if there is corrosion the cable side will vary or be much lower----also check ground cable where it attachs to the frame if loose tghten

Trailblazers are very well known to have problems with the ignition switch. Try jiggling the key and or key off/on a few times to see if anything changes. If this works or seems to make a difference, get you vehicle serviced soon and get the ignition switch replaced - around $150 (parts & labor).