If you weigh more than about 185 lbs, you can pretty well bet that the front end of the bike will be under sprung (need stiffer springs and possibly thicker fork oil). The rear is likely under sprung as well, but that doesn't seem to affect the handling quite as bad unless you carry a lot of luggage or a passenger.
The first thing to do that will make a difference is to set your static sag. It is not hard. There are LOADS of YouTube vids and help pages that will show you how to do it. Having an assistant does help though. Basically, this is adjusting the preload on the springs so that when you sit on the bike, the suspension sits with the suspension compressed about 1/3 of its total stroke length. So if the forks have 9" of travel, they should compress about 3" from their fully extended length when you sit on the bike. It is the same principle for the rear, but the stroke length will be different. You need the assistant because you have to get each wheel off the ground to allow full extension of the shock or fork, measure between a fixed reference on the bike and a point on the suspension, then set it back down, sit on it, and measure between those same reference points again to check the travel. Get this right, and it does make a noticeable difference if it is not set correctly when you start.
Beyond setting your static sag, what you can do is limited by the features of your bike's suspension. Cheap bikes like a KLR 650 have NO adjustment on the forks unless you get inside the forks. More expensive bikes will have compression damping adjustment and maybe preload adjustment (used to adjust static sag), and the best will also have rebound damping adjustment as well. Once your sag is set, you adjust compression and rebound to your taste. Basically, you don't want to have the bike acting like a pogo after you hit a bump, but nor do you want it to be really harsh when it takes a hit. Ideally, the suspension will give enough to absorb an impact without being harsh or collapsing too far (compression damping). Then, once the impact is gone, you don't want the suspensoin to just unload like a rocket but it needs to extend fast enough to be ready for that next impact. You might think that trying to adjust for all that sounds silly, but once you get decent suspension where you can actually feel the difference, it is amazing how much more stable the bike will feel as you experiment with the settings and get it where you like it. This is not really something you need someone to do for you because it is a personal taste thing.
I had a KLR 650 and a Strom 650. Both were way under sprung, even for me at 190 lbs. The front end would dive under braking and the bikes just kind of wallowed through the corners when being ridden even remotely fast. I simply put the right springs and oil in the front for my weight, in addition to getting the sag set right. It cost a few hundred bucks for each bike. It made a world of difference for both bikes. I never messed with the rear because they were good enough for the kind of riding I did on them. You might want to talk to a good suspension guy to figure out what you might want to do to the front and whether or not to have this person do the work for you or just tell you what you need to order if you want to do it yourself.
Messing with the rear is more difficult. Most stock shocks aren't really worth messing with other than perhaps putting a stiffer spring on the shock. That still leaves you with whatever adjustments it has (or doesn't). If you swap the spring, you might need a shop to do it for you just because you have to compress the shock to do it. Really trying to get a good rear shock will start costing a nice chunk of money. Cheap shocks will have a threaded collar preload adjuster or maybe a stepped collar (KLR). This gives some flexibility to adjust for changing loads (adding luggage/passenger). Nicer shocks have a remote preload adjusting knob that you can simply turn one way or the other to add or take out preload as needed. This is basically keeping your static sag set for different loads. They may also have compression/rebound control.
I was riding my VFR 800 two up on some twisty roads that had some nasty whoops in the pavement. When I was pushing a fairly brisk pace, I could feel the bike doing the pogo thing and kind of wallowing through the corners. The front end wasn't feeling planted either. I added rear preload, added in a few clicks of the compression damping, and it totally transformed the handling of the bike. I used to think racers were just prima donnas when they would talk about a half click of this or that, but it really does make a difference.
I now have a BMW 1200 GS with Ohlins on the front and back. It is amazing how well the bike handles when it is set up right with really good suspension. Good suspension isn't always cheap, but it is the best mod you can do to most any bike. I can't recommend anyone in particular in your area though. I would just suggest trying to set your sag and then perhaps spending an afternoon on a lonely road just playing with the settings to get what you like THEN WRITE THEM DOWN! Suspension with clickers may go from 0-30 clicks and you want to remember which click you are one. Figure out how many turns you are in to the range of turns on your remote pre load adjuster on the rear, both when you ride solo and when you ride with luggage, a passenger, or both. It makes doing those quick changes easy. IF you find that your stock suspension simply cannot be adjusted enough, THEN you need to think about spending some money to upgrade the suspension to get the desired adjustability.
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