We're two young North American ecologists without much money who are venturing through Guatemala this Jan/Feb. Our interests are to spend time hiking and looking at flora and fauna while remaining safe and operating within our budget. Hoping to take chicken buses around. Planning on hiking around Nebaj and Acul and then heading to Tikal. What is the best way to get there? From where can we catch a bus, and about how long would it take? Also, is it reasonably safe to hike and possibly camp in forests around Tikal? Any tips much appreciated!
As you will be in Nebaj I also recommend the hike to Cocop and once in Cocop we returned to the main road by following the rio azul trail (ask in the village).
As you will go through Coban on your way to Tikal you might want to take a bus to Lanquin and go to Semuc Champey (amazing scenery, caves, the hike to the mirador is great and the pools and river are beautiful and fun).
So Nebaj to Tikal :
Nebaj to Uspantan (took 90 minutes, we stayed there for a night) and on to Coban (that took 3 hours in 2011, the road was bad because of landslides). From Coban you can take a shuttle but we did it by bus. Coban to Chisec (90 minutes, we stayed there for a while too, nice caves) and on to Sayaxche (2 hours). There you cross the river on a small ferry. On the other side minibuses will be waiting to take you to Santa Elena/Flores. I think it was another 2 hours.
From the Santa Elena bus station you can take a minibus to Tikal (could take 2 hours, depends on how many stops it makes, this is not a tourist shuttle (there are many of those leaving from Flores), cost is Q30).
We also loved Yaxha. Worth a visit for sure.
You can hike a lot at Tikal (in the park itself). You can also camp there. There is a campground close to the gatehouse. They used to provide tents but no longer when we were last there in March 2014. You can put your own tent up or they can provide hammocks to sleep in (with a thick sheet draped around them to protect you from bugs). Near the campground there are a few trails as well.
You can camp at the Tikal campground and it should be perfectly fine to wander around the park during the day, but as a general rule I wouldn't try to camp in the jungle near Tikal itself otherwise--it's a big tourist draw and as such has its share of small-time criminals and petty thieves who sometimes poke around at night (though I haven't heard of anything happening in a while). However, there are tons of other archaeological sites in more remote parts of the Peten where it's perfectly safe to camp, and indeed it's the only real option since many are too far removed for a day-trip. I highly recommend a visit to some of those places if you can swing it: Nakum/Naranjo, Holmul/Cival, Mirador and surrounding sites, etc.
I second the suggestion that you visit Yaxha. Pretty easy to get to and it's one of my favorite sites.
And since you're ecologists, you might want to consider a visit out to El Zotz, not far from Tikal (accessible by tour). Similar forest, with a large bat colony living in the cliffs beneath the site. I haven't been myself but I know people who have worked there and it sounds like a pretty lovely spot.

Thank you, aslukas! Looking at google maps, we see trails through the Tikal national park and beyond, up toward the other spots you're mentioning and toward the border with Belize. We'd like to just use those trails to get around. Do you know anything about those trails, and whether people use them for hiking? Thanks so much!
Since you mention "using those trails to get around" I presume you're talking about trekking, rather than day hikes. Correct?
That area has quite a few jungle "roads" that sort of split the difference between road and trail; they're mostly used by local foresters. While foreigners do occasionally strike out on these routes, it's really not something I'd recommend: they mostly lack sign-posting and it's incredibly easy to get lost since the terrain doesn't have all that many obvious points of reference (and water is scarce in the dry season). Plus, depending on where you go, there is some illegal settlement, poaching and illegal logging, and looting in the remoter parts of the Peten--you don't want to accidentally stumble into any of that. So if you do decide you want to strike out into the forest, I really recommend hiring a guide. There are some great community-run tourism organizations and forestry cooperatives that are worth looking at in that regard, like Bio Itza' (http://www.bioitza.com/about.htm).
The only roads/trails I see on Google Maps are the main roads, which are paved, although the various feeder roads will take you to some wonderful spots (like Yaxha) where the hiking is very good.
Most people will hike those trails with a guide. For example El Mirador is usually a 5 day trek. I think it would be easy to get lost in the jungle and I don't know how safe it is. I am sure people have done it on their own but you would have to really know what you are doing. Tikal to El Zotz is a day hike but most people book a tour, I don't know if you could just find a guide to take you.
It is possible to hike from Yaxha to Nakum. We were told it is a 4 hour hike when we were in Yaxha and you can camp at Nakum (we did not go because we did not have a tent). I believe you can actually hike from Nakum to Tikal but again, I would not attempt that without a guide.
To get to Yaxha from Tikal we took a minibus to El Remate but got off just before town at the cruce. There we waited for a bus but got a ride in an empty tourist van on its way to Belize. He dropped us off at the road leading to Yaxha. Not much traffic. We finally got a ride in a military truck. They dropped us off at El Sombrero. El Sombrero seemed to be the only hotel/lodge close to Yaxha (at least in December 2011), $35 for a double room. If we had had a tent, we could have camped at Yaxha.
And there is actually a bus going from Tikal to Uaxactun (in the afternoon). You can camp there as well.