Some things will require planet-hopping
The easy way to find planets is to simply film all of the starry the starry sky, while traveling along a known trajectory. Eddie — your friendly Sirius Cybernetics shipboard computer — will analyze the movements of all the bright dots, and if any of them move in relation to others, then you know this is a planet. From this it is fairly routine trigonometry for Eddie to find the positions of the different planets.
Keep doing this for a while, say an hour or so and Eddie will keep plotting the positions of the planets. From this Eddie finds out the velocity, that is to say the speed and direction of the planets. With this Eddie will quickly calculate the orbital parameters of the different bodies in the system.
Calculating the size of the different bodies is just a matter to putting your shipboard telescope on them. Using the measured size and the appearance of the planets you can approximate the mass. Eddie can do this for you as well. You will end up within the correct magnitude, even if you may be off by a few hundred percent.
To get an accurate measurement of the mass of the bodies however, you need to get fairly close to them and use your gravimeter. Start with the star itself, then you will have to go planethopping, getting close enough to each so that you can approximate that the other planets' respective gravity will not be significant during your measurements.
And like so you have what you asked for. Share and Enjoy!
After question edit
One thing that is worth to notice is that space is big. Really big. You just won't believe how vastly, hugely mind-bogglingly big it is. I mean, you may think it's a long way down the street to the chemist, but that's just peanuts to space! Listen...
...and so on.
The point is that the risk that you will run head first into a planet is negligible. Sure, Jupiter and Saturn are gigantic balls of obstacle-ness that will do very bad things to your spacecraft should you drive it straight into them, but compared to space itself they are really tiny.
If you wish to mitigate this (non-)risk the answer is still the same: keep filming the sky — preferably in infrared — and be on the lookout for "stars" that move. Objects as huge as Jupiter will be visible well in time for you to take evasive action. And if you wish for even more risk mitigation, send a little remote controlled drone ahead by a few hours, letting that do the job, and follow in its footsteps.