Well it's exciting when anyone decides to take up storm chasing! Compared to UK weather The Plains will likely blow you away (although hopefully not literally). I welcome you Brits out here and wish you luck. I think your success will highly depend on how active the weather pattern is during the 23 day period you choose. If you are unlucky enough to get a year like this one has been with a large ridge much of the time and jet stream far north then good luck to seeing anything much weather wise. If you happen to hit an active pattern with a large negative tilt trough, cold air aloft, moisture, and good jet dynamics then look out! In that case safety would be one of my concerns because you'd need care in knowing what you are about to get yourself in and out of. Some storm systems are very large - perhaps Derechos or MCS. It is much easier to deal with slow moving discrete storms. Storms and systems tend to move much faster in the early season (such as April).
I started out on my own. I just decided to follow a tornado warning of a cell that had recently passed near my house. I went out with few items to assist - just a road map, a video camera, and perhaps a weather radio. First time I didn't see much, and didn't really know what I was looking at. Second time out a day or two later I nailed it and was ground zero to the action - almost part of the action. It was very exciting. Later chases in ensuing years was similar, and perhaps I was lucky at times to not be trapped. I have some 'interesting' memories of these occurrences.

But it's all good.
You probably have low expectations; however I feel confident that primarily if you happen to pick an active period and have the proper equipment you can do well. Being able to forecast is great, but a large percentage of what you gain from doing it yourself can be gotten just by watching SPC's Forecast Products Page. Go by Convective Outlooks, then Mesoscale Discussions, and then Warnings issued by the NWS. Most likely you will want to go to the areas of highest probability and sig severe hatching on the Outlooks. Yep, get an inverter (UK specifications if that is what your notebook supports) and a powerstrip with surge protection so you can plug in a bunch of devices for charging. Make sure you have radar from some decent sources. Even a blob showing a storm in a higher risk area will indicate a reason to intercept. More experienced chasers are more discerning, but under the right conditions you can get into plenty a trouble with such a storm. Use a source for warning information also, and I recommend a navigation / mapping software such as Delorme Street Atlas. All of these things aren't essential but will likely increase your chances of seeing something and increase your safety. PS: Using equipment and radar, etc is useful but make sure you spend plenty of time watching and examining the sky for what is going on. Pay attention to the big picture and be active / proactive in identifying structure and trends. Don't have your head down in the vehicle too long because storm environments are dynamic and can change quickly. For radar, warnings, etc I would recommend GR3 by Gibson Ridge.
As others have said definitely take the time to understand storm structure. There is nothing worse than hearing about tornadoes you missed because you were in the wrong part of the storm. Your approach and intercept to the inflow region and the road network at time of interception is everything. Usually always better to intercept than chase a storm down particularly in early season when storm speeds are faster. While intercepting a storm consider your road options for safety and exits if a bad part of the storm runs you over or you need to avoid a tornado / big hail, etc. Keep in mind really severe storms can have large areas of straight line winds exceeding 80 to 100 mph and may be slinging very large hail. For the big hail you may want eye protection. Try not to get out and run in it.

. Be aware that what you attempt to chase during the day, may chase you at night. Yep use good chase etiquette and understand road rules and safety. Sleeping in your vehicle is probably ok if you are careful. A lot of folks say to not sleep in road side parks, or in particular not certain ones. As others mentioned you can park in truck stop parking lots which are well lit, or at Walmarts. Truck stops provide showers, and so do Campgrounds of America (KOA's). When I sleep in my vehicle I usually just find the most remote off the beaten path dark place I can find where no one will bother me. You will usually know when a car approaches even if you are sleeping. As the Metallica song says... 'sleep with one eye open'.
Finally if you aren't sure where to go, you are welcome to give me a call, but my disclaimer is I take no responsibility for your safety or what you do with the information.