if you are riding essentially flat ground the 125 will be a lot easier to ride and less likely to get you into trouble. When it comes to hill climbs or other situations where you need power a 250 will make up for a lack of skill, but it will also get you into trouble twice as fast.
Like Patman said, you won't "outgrow" a 125 2t. The frame size is exactly the same as the 250 and if you have the skill it has enough power.
I am 6' tall, 190 lbs and I started off on a 125. My experience is that when I was riding technical trails the lighter weight of the 125 was an advantage, I was fairly equal in the typical trail and the only time I really wanted more power was when I hit a fire / pole line road or a long sand wash.
After I started racing enduros I decided that I needed a 250 because they didn't have a 125cc class for age 50+, it was either 450 or 250. Figuring that I was at a disadvantage riding my 125 I "upgraded". Funny thing is that the best finish I have had was on the 125!
There is a strong likelihood that you will eventually want the power of a 250. To start off on one will be a bit harder to learn and you may in fact learn some bad habits but if you can avoid killing yourself the first few rides you will probably be OK.
With a $3K budget you should be able to pick up a fairly decent bike. In my opinion, you are better off finding a bike that was well used but well maintained rather than one that was lightly used and poorly maintained. Two strokes are pretty much all "race" bikes which means that they have high performance engines which means they need a lot of maintenance. Not just oil changes but top end rebuilds. Note that this is true of the 4-stroke race bikes as well.
When looking at a bike ask the seller when the last top end rebuild was done. If it hasn't been done recently then figure that it will need one soon. The chain and sprockets are wear items, how long since they have been changed?
Don't forget that you need to leave some budget for all the safety gear. Do NOT get on a bike without helmet, boots and goggles. I highly recommend a chest protector and gloves. I won't ride without knee pads.
Good luck!
Rod