You don't *need* a breadboard, they're just nice for keeping your circuit organized, clean, and robust. You can certainly dangle components in a drunken spiderweb configuration, and it will still work if the connections are all correct.
What you will be looking for in a resistor, is a run of the mill "axial" resistor. This is the kind that is usually beige in color, has a wire lead sticking out both ends, and has multiple colored "bands" around it. A 4.7k resistor with 5% tolerance (means that the actual resistance will be within 5% of the nominal) will have stripes of these colors (in this and only this order): YELLOW, PURPLE, RED, GOLD
Yellow means "4" and in the 10's place
Purple means "7" and in the 1's place
(so YELLOW-PURPLE means "47")
Red means "x100"
Gold means 5%
There are color code calculators all over everywhere. Websites, applications on google play, etc.
Standard axial resistor will be 1/4 watt, it is way more than enough power for the job and is what I'd recommend. Bigger or smaller than that and the resistor body will also get bigger or smaller -- 1/4 watt resistors are an ideal physical size for hobbyist. Only go with 1/8 W if you are space constrained, and only go with 1/2+W if you need the higher power.
So you connect one of your resistors from 3V3 to SDA, and your second resistor from 3V3 to SCL. Then you connect your peripheral to all the same pins as you would have before, except that now you are driving some more current into the I2C pins.
Now about clock stretching... this is something done by the PERIPHERAL. Basically, if it needs to slow down the rate that data is being sent to it, it will hold the SCL pin at 0V until it is ready for more data. The consequence of this is, of course, that it makes that i2c bus inaccessible to ALL devices that are connected to it, which doesn't matter if there is only one device on the bus.
A very good device when you're running into issues with wire-level data, is a logic analyzer. Especially one with an ANALOG input capability. These will tell you what is *actually* going on, on the wires, so if you see the clock line being held LOW, you know that there is clock stretching going on (both lines will remain "high" while inactive). With analog capability, you will be able to see the *shape* of the curves, which will tell you if you need more or less pull-up current to match the bus. This would be *especially* useful for you, since you are using i2c-gpio instead of hardware i2c. That makes it especially important to know what is actually going on with the wire.
Unfortunately, I can't suggest what logic analyzer to get. I can tell you that Saleae makes *great* devices, but they're hella expensive -- $500 for their "entry level" unit now. I got a Logic 4 for $100, but they've since discontinued it.
Now looking at that module, there **IS AN ALTERNATIVE** to clock stretching i2c. And that alternative is using UART instead of i2c.