If you don't know anything about BGP, it will be hard to diagnose problems. However, if your setup is not that complicated (i.e. you don't use all of the available features) and if you are interested and willing to do some reading (see here for a good introduction) and invest some time, it is not all that hard to do basic debugging without mastering every single aspect of it. This is the same with all technologies.
BGP can get very complicated if you have a large network with many different transit providers and peerings (networks with whom you exchange non-transit traffic). If you start with two routers, two upstreams and have everything installed at the same colocation facility, you will end up with a quite manageable setup which will provide the resilience you are looking for.
As for the kind of problems, a very common pitfall that comes into my mind concerns traffic engineering. While you have full control over what link you send your outbound traffic, the same is not true for inbound traffic. With BGP, you have only indirect means to tell the rest of the world over what path you would like to receive the packets destined for your network.
Knowing this is important for planning your setup. For example, let's suppose you have one 1 Gbps connection to provider A and two 100 Mbps connections to providers B and C and you have around 150 Mbps of inbound traffic distributed over these three lines. Now, suppose that the link to provider A fails and the whole 150 Mbps will come in through B and C. There is no guarantee that the two links will be balanced, it could very well be that one of the links gets saturated and your services would be badly reachable even though you have enough bandwidth and you are even connected to the Internet over two paths. This is because you don't have full control over the inbound traffic. In that case it would have been better to have only two upstream providers, with a 1 Gbps link to each one.