Hi,
As MarkF said, the AD9834 is a very good choice. And besides having amplitude control, it provides a way to cancel offset thanks to its differential output. You can see another project, based on the AD9834, on my blog (it is in Portuguese, but the automatic translation to English is good):
http://www.bloguetronica.com/2018/07/gerador-de-funcoes-gf2.htmlYou can find AD9834 modules cheap on eBay. With a CP2130 evaluation board (also cheap), you can make yourself a decent function generator. You may have to change or even implement a reconstruction filter. Having a 7th order, 0.01dB Chebyshev aligned low pass filter is essential. You can use a lower order one, or instead choose a Butterworth aligned one.
If you wish to have more power, you can choose an AD9102. It has all the features of the previous, but with the addition of direct amplitude and offset control (i.e. inside the chip itself), plus more waveforms at a higher sampling rate.
For the amplifier stage, definitely you'll need an high speed op-amp with a gain–bandwidth product in the order of thousands of MHz, at least, if not in the order of GHz. Choose wisely! My suggestion, you should choose the LMH6612 or similar for the output stage. Take into account that this op-amp will need to drive 50

loads, rail-to-rail if required.
Kind regards, Samuel Lourenço