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You need to amplify the output voltage, but the question must first be asked: 0-10V for what temperature range? The LM35 is a Centigrade sensor with 0V at 0°C and +10mV for every 1°C thereafter. So if you amplify the output by a factor of 10, you'll get 10V at 100°C. If you amplify it by 100, you'll get 10V at 10°C. If you amplify it by 25, you get 10V at 40°C.
You need to amplify the output of the LM35 with a DC amplifier with a gain of 10, like an opamp with the gain set to 10. Since you need to amplify a DC voltage, you will have to either use an opamp with a very low offset voltage (expensive) or use a less exotic opamp and provide a means of nulling the offset.
Yes, and unless you need more than two opamps, you can use the dual-opamp LM358. The specs are practically identical to the LM324. Whichever type you use, you'll need a dual power supply, i.e. a balanced +/- supply so that the output can be set to 0 at 0°C. The power supply also has to be at least ±12V, preferably ±15V since the output voltage has to go up to 10V.
Which ADC are you using? Most of the ADC's works till 5v only so there is no point to go till 10v.
Instead of amplifying the signal you can even use a higher resolution ADC suppose if you use a 12bit ADC you can detect a change of 1.22mv.
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