This paper explores the road to vastly improving the broadband connectivity in future 6G wireless systems. Different categories of use cases are considered, from extreme capacity with peak data rates up to 1 Tbps, to raising the typical data rates by orders-of-magnitude, and supporting broadband connectivity at railway speeds up to 1000 km/h. To achieve these, not only the terrestrial networks will be evolved but they will also be integrated with satellite networks, all facilitating autonomous systems and various interconnected structures. We believe that several categories of enablers at the infrastructure, spectrum, and protocol/algorithmic levels are required to realize the connectivity goals in 6G. At the infrastructure level, we consider ultra-massive MIMO technology (possibly implemented using holographic radio), intelligent reflecting surfaces, user-centric cell-free networking, integrated access and backhaul, and integrated space and terrestrial networks. At the spectrum level, the network must seamlessly utilize sub-6 GHz bands for coverage and spatial multiplexing of many devices, while higher bands will be used for pushing the peak rates of point-to-point links. The latter path will lead to (sub-)Terahertz communications complemented by visible light communications in specific scenarios. At the protocol/algorithmic level, the enablers include improved coding, modulation, and waveforms to achieve lower latency, higher reliability, and reduced complexity. The resource efficiency can be further improved by using various combinations of full-duplex radios, interference management based on rate-splitting, machine-learning based optimization, coded caching, and broadcasting. Finally, the three levels of enablers must be utilized also to provide full-coverage broadband connectivity which must be one of the key outcomes of 6G.