The key word being "budget" if indeed you need to control stuff like 'poison ivy" (which animals can/do eat) heavy under brush, etc then Roundup is not the best buy.
I'll stand by "Farm Works" non-selective weed control concentrate for both effectivness and money. 2.5 gal concentrate at Tractor Supply. The weed/brush killer has the same ingredients as Roundup annnnd there is no need to buy "specially marked bottles" for select weeds like poison ivy.
Carefully follow the instructons as the herbicide is what it says.."NON-SELECTIVE".
For spot or close tolerance weed control place a soup can or styrofoam cup over the nozzle of the spray wand using a course spray. (course spray is generally recommended to minimize overspray in any condition) Placing a cup or can over the nozzle further limits overspray and will allow for spraying a 'weed' mixed in with the flowers. Another trick, if the budget allows, buy a sprayer having a "bubbler or foaming" applicator tip. Lastly but not leasely, for close application, us as little as pressure as possible. Hold the applicator tip over the target with valve open and slowly work the pressure pump.
For ivy growing up and around trees, not a good idea to spray the tree trunk. Cut the vine at ground height and spray any ivy on the ground. Caution, dead ivy vine still contains the nasty oil. The trick is to stop the ivy from spreading.
Under brush, ivy, etc must have leaves plus bushes and such should be minimal height or a second application may be required for effective control.
Glyphosate is the key ingredient for weed control in most weed/brush killers and must be absorbed by the plant where it settles into the roots and prevents the plant from feeding keeping the plant from returning.
Instructions say to apply at least an hour before any rain. Once the product is dry, it's safe & good to go
HOWEVER, don't expect the area to remain weed free as seed and weed not yet sprouting will take over. Stupid weeds.
Remember to read and follow all the directions & recommendations when using any chemicals.
Mowing the area before the weed goes to seed is the next best solution. Of course, that's a constant job but over time, will prove effective.
Crappie-Slayer, I'd like to recommend making any changes to your newly purchased parcel of land [congratulations BTW]. Be subtle with cleaning up and making any changes. Not that it may happen to you but I did something similar and the deer completely avoided the area.
Cutting a trail is one thing but clearing the area caused problems. It's their house and they know, somethings up, when the furniture has been re-arranged.
Before anybody panics about using non-selective herbicides, even near water. Stop by a local grain farmer and ask "what's on your fields"?