You need a OS Map and understanding of what the MoD flags mean. It is not somewhere I’d recommend anyone goes without some planning, even with marked trails, because of how big it is.
If there are raised red flags it’s best avoided The MoD do publish dates in advance, or certainly used to.
Personally I would not want to ride on the plain as there are so many unknowns from hidden soldiers to great bustards wanting to come at you. Even Warminster Saddle club no longer does hacks.
But it might be worth checking out The Old Shaftesbury Drove. It's easy to access just by Salisbury racecourse and goes west for miles.It also goes east towards the city but I think it peters out theses days before you get there. Years back it was the route for sheep heading to London.
I used to live on the edge of Salisbury Plain many years ago and having hunted there a couple of times I would be wary of riding into the plain, we weren't supposed to go into the plain when the red flags were flying but there was a lot of bridleways heading off and onto the plain around the Market Lavington area.
As long as you stay on the perimeter track or on bridleways you're fine.
I've ridden up there for about 10 years and will be moving to a yard where we'll hack straight onto the plain.
Obviously just like with any hacking you need to be prepared for other users on foot , bikes, motorbikes , tractors and you might see some marching troops (although in 10 years only saw those once ). You might hear bangs and see low flying aircraft - but horses stabled near the Plain soon get immune to those.
If the red flags are up it means they're live firing so best to not stray from the perimeter track then.